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IIT JEE Super Course in Mathematics: Coordinate Geometry and Vector Algebra

Trishna Knowledge Systems

Chapter 2

Circles and Conic Sections - all with Video Answers

Educators

Chapter Questions

16:48

Problem 1

(i) Find the equation the circle of radius $\sqrt{73}$ whose centre lies on $x-$ axis and which passes through the point $(2,3)$.
(ii) Find the equation of the circle with two diameters along the lines $x+7 y=23$ and $5 x+2 y=16$ and which passes through the point $(5,7)$.
(iii) Find the length of tangent from the point $(3,-5)$ to the circle $2 x^{2}+2 y^{2}-60 x+12 y+210=0$
(iv) Find the equation of the circle with centre at $(-4,2)$ and touching the line $x-y-3=0$.
(v) Find the equation of the circle, through the intersection of circles $x^{2}+y^{2}+2 x=0$ and $x^{2}+y^{2}-3 y=0$ with its centre on the line $x+2 y-8=0$.

Saad Ali Khan
Saad Ali Khan
Numerade Educator
15:08

Problem 2

(i) Find focus, vertex, directrix and axes of the conic represented by the equation $x^{2}+2 y-3 x+5=0 .$
(ii) Find the equation of the tangent to the parabola $y^{2}=9 x$ which passes through the point $(4,10)$ Also find the point of contact.
(iii) Find the eccentricity, focus, vertices, directrices, length of the latus rectum, and equation of a latus rectum of the conic $x^{2}+4 y^{2}+2 x+16 y+13=0$
(iv) Find the equation of the ellipse whose eccentricity is $\frac{2}{3}$, the latus rectum is 5, whose centre is at the origin and whose major and minor axes are $x$ and $y$ axes.
(v) Find the equation of the hyperbola whose eccentricity is $\sqrt{3}$, focus is at $(1,2)$ and the corresponding directrix is $2 x+y-1=0$

Harshita Goel
Harshita Goel
Numerade Educator
05:11

Problem 3

Find the equations of the circles whose centres lie on the line $4 \mathrm{x}+3 \mathrm{y}-2=0$ and which touch the lines $\mathrm{x}+\mathrm{y}+4=0$ and $7 x-y+4=0$.

Harshita Goel
Harshita Goel
Numerade Educator
03:54

Problem 4

Show that the two circles $x^{2}+y^{2}+2 a x+c=0$ and $x^{2}+y^{2}+2 b y+c=0$ touch if $\frac{1}{a^{2}}+\frac{1}{b^{2}}=\frac{1}{c}$.

Saurabh Chandra
Saurabh Chandra
Numerade Educator
04:27

Problem 5

Find the equation of the circle passing through the point $(0,1)$ and touching the curve $y=x^{2}$ at the point $(2,4)$.

Goutam Chand
Goutam Chand
Numerade Educator
07:07

Problem 6

Let $S$ and $S^{\prime}$ denote the foci, SL the semi latus rectum of an ellipse. LS' produced cuts the ellipse at $P$. Show that the length of the ordinate of $\mathrm{P}$ is $\frac{\left(1-\mathrm{e}^{2}\right)^{2}}{\left(1+3 \mathrm{e}^{2}\right)}$ a where, $2 \mathrm{a}$ is the length of the major axis and $e$ is the eccentricity of the ellipse.

Harshita Goel
Harshita Goel
Numerade Educator
04:34

Problem 7

If the normal to the ellipse $\frac{x^{2}}{14}+\frac{y^{2}}{5}=1$ at the point $\theta$ intersects the curve again at the point $2 \theta$, prove that $\sec \theta=-\frac{3}{2}$.

Harshita Goel
Harshita Goel
Numerade Educator
03:57

Problem 8

If the normal at an end of a latus rectum of an ellipse passes through an extremity of the minor axis, show that the eccentricity e of the ellipse is a solution of the equation $\mathrm{e}^{4}+\mathrm{e}^{2}-1=0$.

Harshita Goel
Harshita Goel
Numerade Educator
03:01

Problem 9

Find the equations of the straight lines which are tangents to the parabola $y^{2}=8 x$ and the hyperbola $3 x^{2}-y^{2}=3 .$

Goutam Chand
Goutam Chand
Numerade Educator
06:08

Problem 10

(i) Show that the conic $12 x^{2}+7 x y-10 y^{2}+13 x+45 y-23=0$ is a hyperbola
(ii) find the coordinates of its centre
(iii) find its asymptotes

Harshita Goel
Harshita Goel
Numerade Educator
01:48

Problem 11

The parametric equation of a point on the circle $x^{2}+y^{2}-8 x-6 y+16=0$ is
(a) $(4+3 \cos \theta, 3+3 \sin \theta)$
(b) $(3 \cos \theta-4,3 \sin \theta-3)$
(c) $(2+3 \cos \theta, 3+2 \sin \theta)$
(d) $(4+9 \cos \theta, 3+9 \sin \theta)$

Goutam Chand
Goutam Chand
Numerade Educator
02:49

Problem 12

Equation of the directrix of the parabola $4 y^{2}+12 x-12 y+39=0$ is
(a) $\mathrm{x}=\frac{-7}{4}$
(b) $y=\frac{3}{4}$
(c) $x-y=0$
(d) $2 x-3 y=0$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:20

Problem 13

If $P$ is any point on the ellipse $4 x^{2}+16 y^{2}=64$ whose foci are $S$ and $S^{1}$, then $S P+S^{1} P$ is
(a) 4
(b) 8
(c) 12
(d) 16

Goutam Chand
Goutam Chand
Numerade Educator
02:32

Problem 14

The normal to the hyperbola $4 \mathrm{y}^{2}-5 \mathrm{x}^{2}=20$ at $(-4,5)$ is
(a) $y-x=9$
(b) $5 y-4 x=41$
(c) $y+x=1$
(d) $x+y=5$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:21

Problem 15

Which of the following equations does not represent a rectangular hyperbola?
(a) $x y=c^{2}$
(b) $x^{2}-y^{2}=a^{2}$
(c) $\frac{y^{2}}{b^{2}}-\frac{x^{2}}{a^{2}}=0$
(d) $x=$ ct, $y=\frac{c}{t}$

Goutam Chand
Goutam Chand
Numerade Educator
04:51

Problem 16

Consider the circles
$C_{1} \equiv x^{2}+y^{2}-4 x+6 y-2=0$
$C_{2} \equiv x^{2}+y^{2}+9 x+6 y-2=0$
Statement 1
The centre of the circle passing through the intersection of the circles $\mathrm{C}_{1}=0$ and $\mathrm{C}_{2}=0$ and through the point $(1,1)$
lies on the line $3 x+y=0$
and
Statement 2 Any circle through the intersection of $\mathrm{C}_{1}=0$ and $\mathrm{C}_{2}=0$ can be written as $\mathrm{C}_{1}+\lambda \mathrm{C}_{2}=0$ where, $\lambda$ is a constant.

Saurabh Chandra
Saurabh Chandra
Numerade Educator
02:31

Problem 17

Statement 1 Let $C$ be a circle intersecting the circles $x^{2}+y^{2}+5 x-9 y+3=0$ and $x^{2}+y^{2}-25=0$ orthogonally. Then, the centre of $C$ lies on the line $5 x-9 y+28=0$ and Statement 2 Centre of any circle intersecting two given circles orthogonally lies on the radical axis of the two circles.

Saurabh Chandra
Saurabh Chandra
Numerade Educator
02:47

Problem 18

Statement 1
The line $\mathrm{x}+2 \mathrm{y}-6=0$ intersects the parabola $\mathrm{x}^{2}-4 \mathrm{x}-8 \mathrm{y}+4=0$ at $\mathrm{M}$ and $\mathrm{N}$. The, tangents at $\mathrm{M}$ and $\mathrm{N}$ to the parabola are at right angles.
and
Statement 2 Tangents at the extremities of a focal chord of a parabola intersect at right angles on the directrix.

Saurabh Chandra
Saurabh Chandra
Numerade Educator
03:31

Problem 19

Consider the ellipse $\mathrm{E}: 16 \mathrm{x}^{2}+96 \mathrm{x}+25 \mathrm{y}^{2}=256$.
Statement 1
Tangent at any point $\mathrm{P}$ on $\mathrm{E}$ is drawn. Let $\mathrm{M}$ denote the foot of the perpendicular from a focus $\mathrm{S}$ of $\mathrm{E}$ to the tangent
at $\mathrm{P}$. Then, $\mathrm{M}$ lies on the circle $\mathrm{x}^{2}+\mathrm{y}^{2}+6 \mathrm{x}-16=0$.
and
Statement 2 The auxiliary circle of the ellipse is $x^{2}+y^{2}+6 x=0$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
03:02

Problem 20

Statement 1 The asymptotes of the hyperbola $x y-2 x+4 y-33=0$ are the lines $x+4=0$ and $y-2=0$.
and
Statement 2
Asymptotes of the rectangular hyperbola $x y=c^{2}$ are $y=0$ and $x=0$.

Saurabh Chandra
Saurabh Chandra
Numerade Educator
02:37

Problem 21

Two members of a coaxal system of circles are $\mathrm{x}^{2}+\mathrm{y}^{2}-6=0$ and $\mathrm{x}^{2}+\mathrm{y}^{2}+4 \mathrm{y}-1=0 .$ The equation of the member of
the system passing through the point $(-1,1)$ is
(a) $9 x^{2}+9 y^{2}+16 y-34=0$
(b) $x^{2}+y^{2}-2 x-3 y-1=0$
(c) $x^{2}+y^{2}-2=0$
(d) None of these

Harshita Goel
Harshita Goel
Numerade Educator
04:48

Problem 22

The limiting points of the coaxal system of circles defined by $x^{2}+y^{2}-6 x-6 y+4=0$ and $x^{2}+y^{2}-2 x-4 y+3=0$ are
(a) $(-1,1),\left(\frac{1}{5}, \frac{-8}{5}\right)$
(b) $(1,-1),\left(\frac{1}{5}, \frac{8}{5}\right)$
(c) $(-1,1),\left(\frac{1}{5}, \frac{8}{5}\right)$
(d) None of these

Harshita Goel
Harshita Goel
Numerade Educator
02:15

Problem 23

$(5,-3)$ and $(2,6)$ are limiting points of a coaxal system of circles. The equation of the radical axis of the system is
(a) $x+3 y-1=0$
(b) $x-2 y-3=0$
(c) $x-3 y+1=0$
(d) $x+y-5=0$

Goutam Chand
Goutam Chand
Numerade Educator
02:17

Problem 24

The range of possible values of the eccentricity e of hyperbola is
(a) $1<\mathrm{e}<\infty$
(b) $\frac{\sqrt{13}}{3}<\mathrm{e}<\infty$
(c) $1<\mathrm{e}<\frac{\sqrt{13}}{3}$
(d) $\frac{\sqrt{13}}{3}<\mathrm{e}<2$

Harshita Goel
Harshita Goel
Numerade Educator
01:38

Problem 25

The value of a for which two vertices of the ellipse coincide with the foci of the hyperbola is
(a) $\sqrt{\frac{3}{2}}$
(b) $\sqrt{5}$
(c) $\sqrt{2}$
(d) 2

Goutam Chand
Goutam Chand
Numerade Educator
02:42

Problem 26

The value of a for which the vertices of the hyperbola coincide with the foci of the ellipse is
(a) $\sqrt{2}$
(b) 2
(c) $\sqrt{3}$
(d) $\sqrt{5}$

Harshita Goel
Harshita Goel
Numerade Educator
04:05

Problem 27

For the circles $x^{2}+y^{2}+6 x-16=0$ and $x^{2}+y^{2}-2 x+6 y-6=0$
(a) Number of common tangents is 2
(b) Number of common tangents is 3
(c) Distance of the point $(1,2)$ from the radical axis of the circles is $\frac{7}{5}$
(d) the circles touch internally.

Harshita Goel
Harshita Goel
Numerade Educator
02:03

Problem 28

Consider the parabola $4 x^{2}-4 x+8 y+17=0$
(a) latus rectum of the parabola is 2
(b) no part of the curve lies above the line $y+2=0$
(c) focus of the parabola is at $\left(\frac{1}{2}, \frac{-3}{2}\right)$
(d) focus of the parabola is at $\left(\frac{1}{2}, \frac{-5}{2}\right)$

Harshita Goel
Harshita Goel
Numerade Educator
07:40

Problem 29

For the hyperbola $4 \mathrm{x}^{2}-25 \mathrm{y}^{2}+24 \mathrm{x}-100 \mathrm{y}-164=0$, which of the following is/are true?
(a) eccentricity $=\frac{\sqrt{29}}{5}$
(b) length of latus rectum $=\frac{8}{5}$
(c) Equation of the pair of the asymptotes is $4 x^{2}-25 y^{2}+24 x-100 y-64=0$
(d) The asymptotes are inclined to each other at $\frac{1}{2} \tan ^{-1}\left(\frac{20}{21}\right)$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
02:25

Problem 30

Match the elements of Column I to elements of Column II. There can be single or multiple matches.
Column I
(a) $2 x^{2}+5 x y+2 y^{2}-11 x-7 y-4=0$ represents a/an
(b) The curve $x=3($ cost $+$ sint $) y=4($ cost $-$ sint $)$ represents a/an
(c) $x^{2}-2 x y+y^{2}-26 x-22 y+25=0$ represents a/an
(d) $3 x^{2}-11 x y+10 y^{2}-7 x+13 y+4=0$ represents a/an
Column II
(p) Parabola
(q) Pair of straight lines
(r) Ellipse
(s) Hyperbola

Dwijendra Rao
Dwijendra Rao
Numerade Educator
01:43

Problem 31

The equation of a circle which passes through the origin and cuts off intercepts 'a' and 'b' on the axes is
(a) $x^{2}+y^{2}=a b$
(b) $x^{2}+y^{2}-a x-b y=0$
(c) $x^{2}+y^{2}=a+b$
(d) $x^{2}+y^{2}-2 a x-2 b y=0$

Goutam Chand
Goutam Chand
Numerade Educator
01:37

Problem 32

The locus of the centre of the circle which passes through $(0, a)$ and $(0,-a)$ is
(a) $x+y=0$
(b) $x=y$
(c) $\mathrm{y}=0$
(d) $x=0$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:40

Problem 33

The line $y=m x+c$ and the circle $x^{2}+y^{2}=a^{2}$ do not intersect if
(a) $a=\frac{|c|}{\sqrt{1+m^{2}}}$
(b) $\mathrm{a}<\frac{|\mathrm{c}|}{\sqrt{1+\mathrm{m}^{2}}}$
(c) $a>\frac{|c|}{\sqrt{1+m^{2}}}$
(d) $a c=\sqrt{1+m^{2}}$

Goutam Chand
Goutam Chand
Numerade Educator
01:33

Problem 34

The length of the tangent from any point on the circle $x^{2}+y^{2}-8 x+5 y-4=0$ to the circle $x^{2}+y^{2}-8 x+5 y=0$ is
(a) 1
(b) 2
(c) 4
(d) 6

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:31

Problem 35

The angle between the radical axis and the line joining the centres of the circles $x^{2}+y^{2}+2 g x+2 f y+c=0$ and $x^{2}+y^{2}$ $+2 g_{1} x+2 f_{1} y+c_{1}=0$ is
(a) $\frac{\pi}{3}$
(b) $\frac{\pi}{2}$
(c) 0
(d) $\frac{\pi}{6}$

Goutam Chand
Goutam Chand
Numerade Educator
02:19

Problem 36

The equation of the circle two of whose diameters are along $2 \mathrm{x}-3 \mathrm{y}+12=0$ and $\mathrm{x}+4 \mathrm{y}-5=0$ and which has area 154 square units $\left(\pi=\frac{22}{7}\right)$ is
(a) $(x+3)^{2}+(y-2)^{2}=49$
(b) $(x-2)^{2}+(y+3)^{2}=4$
(c) $x^{2}+y^{2}=154$
(d) $x^{2}+y^{2}=1$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
02:04

Problem 37

The equations $x=a \cos \theta+b \sin \theta$ and $y=a \sin \theta-b \cos \theta$ represent
(a) a circle
(b) a parabola
(c) a line
(d) an ellipse

Goutam Chand
Goutam Chand
Numerade Educator
04:15

Problem 38

The distances from the origin to the centres of the circles $x^{2}+y^{2}+2 k_{1} x-c^{2}=0 ; i=1,2,3$ are in G.P. Then the lengths of tangents drawn from any point on the circle $x^{2}+y^{2}=c^{2}$ to these circles are in
(a) A.P
(b) G.P
(c) H.P
(d) $\mathrm{AGP}$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
02:37

Problem 39

The length of the common chord of the circles $x^{2}+y^{2}+6 x=0, x^{2}+y^{2}+3 y=0$ is
(a) $\frac{6}{\sqrt{5}}$
(b) $\frac{\sqrt{6}}{10}$
(c) $\frac{6}{10}$
(d) $\frac{3}{\sqrt{5}}$

Goutam Chand
Goutam Chand
Numerade Educator
01:59

Problem 40

Area of the circle in which a chord of length $2 \sqrt{3}$ makes an angle $2 \frac{\pi}{3}$ at the centre is
(a) $4 \pi$
(b) $\pi^{2}$
(c) $\frac{1}{2}$
(d) $\sqrt{2} \pi$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
02:22

Problem 41

If the lengths of the tangents from $(3,5)$ to the circles $x^{2}+y^{2}-4 x+4 y-1=0$ and $x^{2}+y^{2}-6 x+10 y-c=0$ are in the ratio $1: 3$, then the value of $\mathrm{c}$ is
(a) $-77$
(b) $-11$
(c) $-303$
(d) $-77$

Goutam Chand
Goutam Chand
Numerade Educator
01:53

Problem 42

Tangents drawn to $x^{2}+y^{2}=16$ from $P(0, h)$ meet the $x$ -axis at $A$ and $B$. The value of $h$ for which the area of triangle $\mathrm{PAB}$ is minimum is
(a) 32
(b) $\sqrt{30}$
(c) $\sqrt{31}$
(d) $4 \sqrt{2}$

Shima Shaw
Shima Shaw
Numerade Educator
03:32

Problem 43

The number of points of the form $(a+1, \sqrt{3} a), a \in Z$ lying inside the region bounded by $x^{2}+y^{2}-2 x-3=0$, and $x^{2}$ $+y^{2}-2 x-15=0$ are
(a) 1
(b) 2
(c) 0
(d) 3

Saurabh Chandra
Saurabh Chandra
Numerade Educator
04:31

Problem 44

If the two circles $x^{2}+y^{2}+2 g_{1} x+2 f_{1} y=0$ and $x^{2}+y^{2}+2 g_{2} x+2 f_{2} y=0$ touch each other, then
(a) $\mathrm{f}_{1} \mathrm{f}_{2}=\mathrm{g}_{1} \mathrm{~g}_{2}$
(b) $\mathrm{f}_{1} \mathrm{~g}_{1}=\mathrm{f}_{2} \mathrm{~g}_{2}$
(c) $\mathrm{f}_{1} \mathrm{~g}_{2}=\mathrm{f}_{2} \mathrm{~g}_{1}$
(d) $\mathrm{f}_{1} \mathrm{~g}_{2}{ }^{2}=\mathrm{f}_{2} \mathrm{~g}_{1}{ }^{2}$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:38

Problem 45

For any $\lambda \in \mathrm{R}, \mathrm{x}^{2}+\mathrm{y}^{2}-2 \lambda \mathrm{x}-2 \lambda \mathrm{y}+\lambda^{2}=0$ touches the $\operatorname{line}(\mathrm{s})$
(a) $x^{2}=y^{2}$
(b) $x=y$
(c) $x+y=0$
(d) $x y=0$

Goutam Chand
Goutam Chand
Numerade Educator
03:22

Problem 46

Four distinct points $(0,0),(0,1),(1,0),(2 \mathrm{k}, 3 \mathrm{k})$ lie on a circle. Then the value of $\mathrm{k}$ is
(a) $\frac{13}{5}$
(b) $\frac{5}{13}$
(c) 1
(d) $\frac{1}{2}$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:15

Problem 47

The vertex of the parabola $y^{2}-8 y-x+19=0$ is at
(a) $(3,3)$
(b) $(4,4)$
(c) $(3,4)$
(d) $(4,3)$

Goutam Chand
Goutam Chand
Numerade Educator
02:55

Problem 48

A double ordinate of the parabola $y^{2}=4 a x$ is of length 8 a. Then the angle between the lines from the vertex to its ends is
(a) $\frac{\pi}{3}$
(b) $\frac{\pi}{2}$
(c) $\frac{\pi}{6}$
(d) $\pi$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:22

Problem 49

The equation of a parabola is $(y-k)^{2}=4 a(x-h)$ where, $a>0$. The vertex and focus are respectively at
(a) $(\mathrm{h}, \mathrm{k})$ and $(\mathrm{a}, \mathrm{k})$
(b) $(\mathrm{h}+\mathrm{a}, \mathrm{k}+\mathrm{a})$ and $(\mathrm{h}, \mathrm{k})$
(c) $(\mathrm{h}, \mathrm{k})$ and $(\mathrm{h}-\mathrm{a}, \mathrm{k})$
(d) $(\mathrm{h}, \mathrm{k})$ and $(\mathrm{h}+\mathrm{a}, \mathrm{k})$

Goutam Chand
Goutam Chand
Numerade Educator
01:22

Problem 50

If the focus of the parabola $(\mathrm{y}-\mathrm{k})^{2}=4(\mathrm{x}-\mathrm{h})$ always lies between the lines $\mathrm{x}+\mathrm{y}=1$ and $\mathrm{x}+\mathrm{y}=3$, then
(a) $\mathrm{h}+\mathrm{k}<0$ or $\mathrm{h}+\mathrm{k}>2$
(b) $0<\mathrm{h}+\mathrm{k}<2$
(c) $\mathrm{h}+\mathrm{k}<-2$
(d) $0<|\mathrm{h}-\mathrm{k}|<2$

Samantha Lucero
Samantha Lucero
Numerade Educator
01:11

Problem 51

A normal to the parabola $y^{2}=4 a x$ with slope $^{\prime} m^{3}$ touches the hyperbola $x^{2}-y^{2}=a^{2}$ if
(a) $\mathrm{m}^{6}-4 \mathrm{~m}^{4}+3 \mathrm{~m}^{2}+1=0$
(b) $m^{6}-4 m^{4}-3 m^{2}+1=0$
(c) $\mathrm{m}^{6}+4 \mathrm{~m}^{4}-3 \mathrm{~m}^{2}+1=0$
(d) $m^{6}+4 m^{4}+3 m^{2}+1=0$

Gaurav Kalra
Gaurav Kalra
Numerade Educator
01:54

Problem 52

The circle $x^{2}+y^{2}+2 \lambda x=0, \lambda \in R$ touches the parabola $y^{2}=4 x$ externally only if
(a) $\lambda>0$
(b) $\lambda<0$
(c) $\lambda>1$
(d) $\lambda=1$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:52

Problem 53

If the line $x-2=0$ is the directrix of the parabola $y^{2}-k x-4=0$, then one of the values of $k$ is
(a) 4
(b) $-4$
(c) 3
(d) 2

Goutam Chand
Goutam Chand
Numerade Educator
02:35

Problem 54

The focal distance of a point on the parabola $\mathrm{y}^{2}=12 \mathrm{x}$ is 4 . Then the abscissa of this point is
(a) 1
(b) 3
(c) 4
(d) 6

Saurabh Chandra
Saurabh Chandra
Numerade Educator
02:28

Problem 55

An equilateral triangle is inscribed in the parabola $\mathrm{y}^{2}=4 \mathrm{ax}$ such that one of the vertices is the vertex of the parabola. Then the length of each side is
(a) $\mathrm{a} \sqrt{3}$
(b) $8 \mathrm{a} \sqrt{3}$
(c) $4 \mathrm{a} \sqrt{3}$
(d) $2 \mathrm{a} \sqrt{3}$

Goutam Chand
Goutam Chand
Numerade Educator
01:24

Problem 56

If $\mathrm{y}_{1}, \mathrm{y}_{2}, \mathrm{y}_{3}$ are the ordinates of the vertices of a triangle inscribed in the parabola $\mathrm{y}^{2}=4 \mathrm{ax}(\mathrm{a}>0)$, then its area is
(a) $\frac{1}{8 \mathrm{a}}\left|\mathrm{y}_{1} \mathrm{y}_{2} \mathrm{y}_{3}\right|$
(b) $\frac{1}{8 \mathrm{a}}\left|\left(\mathrm{y}_{1}-\mathrm{y}_{2}\right)\left(\mathrm{y}_{2}-\mathrm{y}_{3}\right)\left(\mathrm{y}_{3}-\mathrm{y}_{1}\right)\right|$
(c) $\frac{1}{8 \mathrm{a}}\left|\left(\mathrm{y}_{1}+\mathrm{y}_{2}+\mathrm{y}_{3}\right)\right|$
(d) $8 a\left|\left(y_{1}-y_{2}\right)\left(y_{2}-y_{3}\right)\left(y_{3}-y_{1}\right)\right|$

Vishal Parmar
Vishal Parmar
Numerade Educator
02:54

Problem 57

The angle between the tangents at the extremities of any focal chord of a parabola $y^{2}=4 a x$ is
(a) $\frac{\pi}{2}$
(b) $\frac{\pi}{3}$
(c) $\frac{\pi}{4}$
(d) $\frac{\pi}{6}$

Goutam Chand
Goutam Chand
Numerade Educator
01:43

Problem 58

The latus rectum of an ellipse is equal to half the length of its minor axis. Its eccentricity is
(a) $\frac{1}{\sqrt{2}}$
(b) $\frac{\sqrt{3}}{\sqrt{2}}$
(c) $\frac{\sqrt{3}}{2}$
(d) $\frac{1}{2}$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:18

Problem 59

The auxiliary circle of the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{16}=1$ is
(a) $x^{2}+y^{2}=25$
(b) $x^{2}+y^{2}=16$
(c) $x^{2}+y^{2}=9$
(d) $x^{2}+y^{2}=4$

Goutam Chand
Goutam Chand
Numerade Educator
01:47

Problem 60

The line $\mathrm{y}=2 \mathrm{x}+\mathrm{k}$ touches the ellipse $4 \mathrm{x}^{2}+\mathrm{y}^{2}=8$ if $\mathrm{k}=$
(a) $\pm 2$
(b) $\pm 4$
(c) $\pm 2 \sqrt{2}$
(d) $\sqrt{2}$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:32

Problem 61

Sum of the focal distances of a point on an ellipse whose major and minor axes are of lengths $2 \mathrm{a}$ and $2 \mathrm{~b}$ respectively, is equal to
(a) 2a
(b) $2 \mathrm{~b}$
(c) $2 \mathrm{ae}$
(d) $2 \mathrm{ab}$

Goutam Chand
Goutam Chand
Numerade Educator
01:23

Problem 62

The sum of the squares of the eccentricities of the conics $\frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ and $\frac{x^{2}}{4}-\frac{y^{2}}{3}=1$ is
(a) 2
(b) $\sqrt{2}$
(c) $\frac{\sqrt{7}}{\sqrt{3}}$
(d) $\sqrt{7}$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
02:08

Problem 63

The distance between the foci of the ellipse $20 \mathrm{x}^{2}+4 \mathrm{y}^{2}=5$ is
(a) 1
(b) 2
(c) 4
(d) 3

Goutam Chand
Goutam Chand
Numerade Educator
02:39

Problem 64

The curve represented by $x=3(\cos t+\sin t), y=4(\cos t-\sin t)$ is
(a) an ellipse
(b) a circle
(c) a hyperbola
(d) a parabola

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:23

Problem 65

The equation $\frac{x^{2}}{r-6}+\frac{y^{2}}{3-r}+1=0$ represents an ellipse if and only if
(a) $\mathrm{r}>3$
(b) $\mathrm{r}>6$
(c) $3<\mathrm{r}<6$
(d) $\sqrt{3}>\mathrm{r}>\sqrt{6}$

Goutam Chand
Goutam Chand
Numerade Educator
01:41

Problem 66

The centre of the ellipse $\frac{(x+y-4)^{2}}{16}+\frac{(x-y)^{2}}{4}=1$ is
(a) $(0,0)$
(b) $(2,2)$
(c) $(1,1)$
(d) $(1,2)$

Goutam Chand
Goutam Chand
Numerade Educator
02:01

Problem 67

The equation of the hyperbola whose transverse and conjugate axes coincide with the $x$ and $y$ axes respectively and which passes through the point $(5,2)$ and whose transverse axis is twice the conjugate axis is
(a) $x^{2}-4 y^{2}=9$
(b) $4 x^{2}-y^{2}=96$
(c) $x^{2}-4 y^{2}=100$
(d) $x^{2}-4 y^{2}=21$

Goutam Chand
Goutam Chand
Numerade Educator
04:09

Problem 68

The line $x \cos \alpha+y \sin \alpha=P$ touches the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ if
(a) $\mathrm{P}^{2}=\mathrm{a}^{2} \cos ^{2} \alpha+\mathrm{b}^{2} \sin ^{2} \alpha$
(b) $\mathrm{P}^{2}=\mathrm{b}^{2} \cos ^{2} \alpha+\mathrm{a}^{2} \sin ^{2} \alpha$
(c) $\mathrm{P}^{2}=\mathrm{a}^{2} \cos ^{2} \alpha-\mathrm{b}^{2} \sin ^{2} \alpha$
(d) $\mathrm{P}^{2}=\mathrm{b}^{2} \cos ^{2} \alpha-\mathrm{a}^{2} \operatorname{ain}^{2} \alpha$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
00:32

Problem 69

The equation of the ellipse with eccentricity $\frac{1}{\sqrt{2}}$ circumscribing the rectangle whose sides are given by $\mathrm{x}=\pm 2, \mathrm{y}=$ $\pm 1$, is
(a) $2 x^{2}+3 y^{2}=11$
(b) $x^{2}+2 y^{2}=6$
(c) $3 x^{2}+2 y^{2}=14$
(d) $2 x^{2}+y^{2}=9$

Himanshu Kushwaha
Himanshu Kushwaha
Numerade Educator
02:05

Problem 70

The eccentricity and the coordinates of a focus of the conic $(5 x-10)^{2}+(5 y+15)^{2}=(x-2 y+7)^{2}$ are given by
(a) $\sqrt{5},(2,-3)$
(b) $\frac{1}{\sqrt{5}},(2,-3)$
(c) $\frac{1}{\sqrt{5}},(3,-2)$
(d) $\frac{1}{\sqrt{5}},(3,2)$

Goutam Chand
Goutam Chand
Numerade Educator
01:37

Problem 71

If the foci of the ellipse $\frac{\mathrm{x}^{2}}{\mathrm{k}^{2} \mathrm{a}^{2}}+\frac{\mathrm{y}^{2}}{\mathrm{a}^{2}}=1$ and hyperbola $\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}-\frac{\mathrm{y}^{2}}{\mathrm{a}^{2}}=1$ coincide, then a value of $\mathrm{k}$ is
(a) $\sqrt{2}$
(b) 1
(c) $\sqrt{3}$
(d) 2

Goutam Chand
Goutam Chand
Numerade Educator
02:29

Problem 72

If $^{\prime} \theta$ ' is the angle between the asymptotes of the hyperbola $\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}-\frac{\mathrm{y}^{2}}{\mathrm{~b}^{2}}=1$, then $\cos \frac{\theta}{2}$ is
(a) $\sqrt{\mathrm{e}}$
(b) $\frac{1}{1+\mathrm{e}}$
(c) $\frac{1}{\sqrt{e}}$
(d) $\frac{1}{\mathrm{e}}$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:46

Problem 73

The number of points from where one can draw mutually perpendicular tangents to the hyperbola $(x-1)^{2}-(y-2)^{2}$ $=4$ is
(a) 1
(b) 2
(c) 4
(d) infinite

Goutam Chand
Goutam Chand
Numerade Educator
02:44

Problem 74

If $\mathrm{AB}$ is a double ordinate of the hyperbola $\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}-\frac{\mathrm{y}^{2}}{\mathrm{~b}^{2}}=1$ such that $\triangle \mathrm{OAB}$ is an equilateral triangle, where $\mathrm{O}$ is the centre of the hyperbola, then the eccentricity e of the hyperbola satisfies
(a) $1<\mathrm{e}<\frac{2}{\sqrt{3}}$
(b) $\mathrm{e}<\frac{1}{\sqrt{3}}$
(c) e $>\frac{2}{\sqrt{3}}$
(d) $\mathrm{e}=\frac{2}{\sqrt{3}}$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
02:53

Problem 75

If the tangent to the ellipse $x^{2}+4 y^{2}=16$ at the point with eccentric angle $\phi$ is a normal to the curve $x^{2}+y^{2}-8 x-4 y$ $=0$, then $\phi$ equals
(a) $\frac{\pi}{3}$
(b) $\frac{-\pi}{3}$
(c) $\frac{\pi}{6}$
(d) $\frac{\pi}{2}$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
02:07

Problem 76

The minimum distance between the parabolas $\mathrm{y}^{2}-4 \mathrm{x}-8 \mathrm{y}+40=0$ and $x^{2}-8 x-4 y+40=0$ is
(a) $\sqrt{3}$
(b) $\sqrt{2}$
(c) 1
(d) 2

Hast Aggarwal
Hast Aggarwal
Numerade Educator
02:52

Problem 77

A line with slope $\mathrm{m}$ touches both $\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}+\frac{\mathrm{y}^{2}}{\mathrm{~b}^{2}}=1$ and $\mathrm{x}^{2}+\mathrm{y}^{2}=\mathrm{c}^{2}$ then
(a) $b<c<a$
(b) $c<b<a$
(c) $\mathrm{b}<\mathrm{a}<\mathrm{c}$
(d) $c \in \mathrm{R} ; \mathrm{a}>\mathrm{b}$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
06:13

Problem 78

The tangent at a point on $16 \mathrm{x}^{2}+25 \mathrm{y}^{2}=400$ meets the tangents at the ends of the major axis at $\mathrm{T}_{1}$ and $\mathrm{T}_{2} .$ Then the equation of the circle with on $\mathrm{T}_{1} \mathrm{~T}_{2}$ as diameter passes through the point
(a) $(1,0)$
(b) $(3,0)$
(c) $(0,3)$
(d) $(0,-3)$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
07:40

Problem 79

The eccentricity of the hyperbola $\frac{x}{a^{2}}-\frac{y}{b^{2}}=1$ with latus rectum $\frac{32 \sqrt{2}}{5}$ and passing through the point of intersection of $7 x+13 y-87=0$ and $5 x-8 y+7=0$ is
(a) $\sqrt{\frac{57}{32}}$
(b) $\sqrt{\frac{57}{25}}$
(c) $\sqrt{\frac{27}{21}}$
(d) $\sqrt{2}$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:21

Problem 80

If the curves $x^{2}+y^{2}-8|x|-9=0$ and $y=|x|+c$ do not intersect, then
(a) $c<-4-5 \sqrt{2}$ or $c>-4+5 \sqrt{2}$
(b) $-4-5 \sqrt{2}<\mathrm{c}<-4+5 \sqrt{2}$
(c) $-5 \sqrt{2}<\mathrm{c}<5 \sqrt{2}$
(d) $-4<c<4$

Dilip Paruchuri
Dilip Paruchuri
Numerade Educator
01:15

Problem 81

If $4 x^{2}+4 y^{2}+4 h x y+16 x+32 y+10=0$ represents a circle, then $h$ is
(a) 5
(b) 0
(c) $-2$
(d) $-5$

Goutam Chand
Goutam Chand
Numerade Educator
01:30

Problem 82

The length of the diameter of the circle which has the lines $12 x+5 y+52=0$ and $12 x+5 y=0$ as its tangents is
(a) 5
(b) 4
(c) 42
(d) 1

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:28

Problem 83

If the circle $x^{2}+y^{2}+2 g x+2 f y+c=0$ touches the $y$ -axis, then
(a) $g^{2}=\mathrm{f}^{2}$
(b) $\mathrm{f}^{2}=c$
(c) $\mathrm{f}^{2}+\mathrm{c}=0$
(d) $\mathrm{f}^{2}=\mathrm{g}^{2} \mathrm{c}$

Goutam Chand
Goutam Chand
Numerade Educator
02:09

Problem 84

If the line $3 x+4 y+20=0$ touches the circle $x^{2}+y^{2}=16$, then the point of contact is
(a) $\left(\frac{-12}{5}, \frac{-16}{5}\right)$
(b) $\left(\frac{12}{5}, \frac{16}{5}\right)$
(c) $\left(0, \frac{-16}{5}\right)$
(d) $\left(\frac{12}{5}, 0\right)$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:08

Problem 85

The latus rectum of the parabola $y=x^{2}-2 x+3$ is
(a) 1
(b) 4
(c) $\frac{1}{4}$
(d) $\frac{1}{2}$

Goutam Chand
Goutam Chand
Numerade Educator
02:13

Problem 86

Two perpendicular tangents to $\mathrm{y}^{2}=4 \mathrm{ax}$ where $\mathrm{a}>0$ always intersect on the line
(a) $x+a=0$
(b) $x+4 a=0$
(c) $x=a$
(d) $x-4 a=0$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:24

Problem 87

If one end of a diameter of the ellipse $4 x^{2}+y^{2}=64$ is at $(2 \sqrt{3}, 4)$, then the other end is at
(a) $(-2 \sqrt{3},-4)$
(b) $(2 \sqrt{3},-4)$
(c) $(-2 \sqrt{3}, 4)$
(d) $(2 \sqrt{3}, 4)$

Goutam Chand
Goutam Chand
Numerade Educator
02:35

Problem 88

The angle between the rectangular hyperbolas $x y=6$ and $x^{2}-y^{2}=5$ at $(3,2)$ is
(a) $0^{\circ}$
(b) $90^{\circ}$
(c) $45^{\circ}$
(d) $60^{\circ}$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:59

Problem 89

The centres of the three circles $x^{2}+y^{2}-4 x-6 y-12=0, x^{2}+y^{2}+2 x+4 y-10=0, x^{2}+y^{2}-10 x-16 y-1=0$ are
(a) collinear
(b) non-collinear
(c) collinear and lie on $\mathrm{x}$ -axis
(d) collinear and lie on $\mathrm{y}=\mathrm{x}+1$

Goutam Chand
Goutam Chand
Numerade Educator
02:36

Problem 90

The radii of the circles $x^{2}+y^{2}=1, x^{2}+y^{2}-2 x-6 y-6=0$ and $x^{2}+y^{2}-4 x-12 y-9=0$ are in
(a) G.P
(b) A.P
(c) H.P
(d) AGP

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:57

Problem 91

The parametric equation of the circle $x^{2}+y^{2}+p x+p y=0$ is
(a) $\mathrm{x}=\frac{-\mathrm{p}}{2}, \mathrm{y}=\frac{-\mathrm{p}}{2}$
(b) $\mathrm{x}=\frac{\mathrm{p}}{2} \cos \theta, \mathrm{y}=\frac{\mathrm{p}}{2} \sin \theta$
(c) $x=\frac{-p}{2}+\frac{p}{\sqrt{2}} \cos \theta, y=\frac{-p}{2}+\frac{p}{\sqrt{2}} \sin \theta$
(d) $\mathrm{x}=\frac{\mathrm{p}}{\sqrt{2}} \cos \theta, \mathrm{y}=\frac{\mathrm{p}}{\sqrt{2}} \sin \theta$

Goutam Chand
Goutam Chand
Numerade Educator
01:40

Problem 92

Diameter of the circle passing through $(2,4)$ and whose centre is at the point of intersection of lines $x+2 y-5=0$ and $x+3 y-6=0$ is
(a) $\sqrt{10}$
(b) 10
(c) $2 \sqrt{10}$
(d) 20

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:08

Problem 93

The equation of the tangent to the circle $x^{2}+y^{2}=25$, at $(3,4)$ is
(a) $3 x+4 y=25$
(b) $4 x+3 y=16$
(c) $x+y=10$
(d) $3 x-4 y=20$

Goutam Chand
Goutam Chand
Numerade Educator
03:24

Problem 94

If the lines $a x+b y+c_{1}=0$ and $a x+b y+c_{2}=0,\left(c_{1} c_{2} \neq 0\right)$ intersect the co-ordinate axes at con-cyclic points, then
(a) $a=b$
(b) $a^{2}=b^{2}$
(c) $\mathrm{ac}_{1}=\mathrm{bc}_{2}$
(d) $\mathrm{ac}_{1}+\mathrm{bc}_{2}=0$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:36

Problem 95

The angle between the tangents to the circle $x^{2}+y^{2}=169$ at $(5,12)$ and at $(12,-5)$ is
(a) $\frac{\pi}{3}$
(b) $\frac{\pi}{4}$
(c) $\frac{\pi}{2}$
(d) $\frac{\pi}{6}$

Goutam Chand
Goutam Chand
Numerade Educator
03:11

Problem 96

The length of the intercept cut off from the line $y=m x+c$ by the circle $x^{2}+y^{2}=a^{2}$ is
(a) $\sqrt{a^{2}\left(1+m^{2}\right)-c^{2}}$
(b) $\frac{\sqrt{a^{2}\left(1+m^{2}\right)-c^{2}}}{\sqrt{1+m^{2}}}$
(c) $\frac{2 \sqrt{a^{2}\left(1+m^{2}\right)-c^{2}}}{\sqrt{1+m^{2}}}$
(d) $\mathrm{a}^{2}\left(1+\mathrm{m}^{2}\right)=\mathrm{c}^{2}$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:41

Problem 97

The angle subtended by the common chord of the circles $x^{2}+y^{2}-4 x-4 y=0$ and $x^{2}+y^{2}=16$ at the origin is
(a) $\frac{\pi}{2}$
(b) $\frac{\pi}{3}$
(c) $\frac{\pi}{4}$
(d) $\frac{\pi}{6}$

Goutam Chand
Goutam Chand
Numerade Educator
02:23

Problem 98

The shortest distance from the point $(4,2)$ to the circle $x^{2}+y^{2}-10 x-4 y-7=0$ is
(a) 1
(b) 5
(c) 6
(d) 11

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:48

Problem 99

If $(p, q)$ is the mid-point of a chord of the parabola $y^{2}=4 x$, passing through its vertex then
(a) $\mathrm{p}^{2}=2 \mathrm{q}$
(b) $q^{2}=2 p$
(c) $\mathrm{p}^{2}=\mathrm{p}+\mathrm{q}$
(d) $q^{2}=p-q$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
02:09

Problem 100

The angle of intersection of the circles $x^{2}+y^{2}+2 a x=0$ and $x^{2}+y^{2}+2 b y=0$ at $(0,0)$ is
(a) $\frac{\pi}{3}$
(b) $\frac{\pi}{4}$
(c) 0
(d) $\frac{\pi}{2}$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
02:46

Problem 101

If the curves $a_{1} x^{2}+2 h_{1} x y+b_{1} y^{2}+2 g_{1} x+2 f_{1} y+c_{1}=0$ and $a_{2} x^{2}+2 h_{2} x y+b_{2} y^{2}+2 g_{2} x+2 f_{2} y+c_{2}=0$ intersect at four
concyclic points, then
(a) $a_{1}-b_{1}=a_{2}-b_{2}$
(b) $\mathrm{h}_{1}\left(\mathrm{a}_{1}-\mathrm{b}_{1}\right)=\mathrm{h}_{2}\left(\mathrm{a}_{2}-\mathrm{b}_{2}\right)$
(c) $\mathrm{h}_{1}=\mathrm{h}_{2}$ and $\mathrm{a}_{1} \mathrm{~b}_{2}=\mathrm{a}_{2} \mathrm{~b}_{1}$
(d) $\mathrm{h}_{2}\left(\mathrm{a}_{1}-\mathrm{b}_{1}\right)=\mathrm{h}_{1}\left(\mathrm{a}_{2}-\mathrm{b}_{2}\right)$

Harshita Goel
Harshita Goel
Numerade Educator
03:05

Problem 102

One of the common tangents of $\mathrm{x}^{2}+\mathrm{y}^{2}=\frac{\mathrm{a}^{2}}{2}$ and $\mathrm{y}^{2}=4 \mathrm{ax}$ is
(a) $y=x+a$
(b) $y=-x+a$
(c) $x=y+a$
(d) $x+y=2 a$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:41

Problem 103

The centre of the ellipse $\frac{(x+y-2)^{2}}{9}+\frac{(x-y)^{2}}{16}=1$ is at
(a) $(1,1)$
(b) $(2,0)$
(c) $(0,2)$
(d) $(0,0)$

Goutam Chand
Goutam Chand
Numerade Educator
03:26

Problem 104

The point of intersection of the curves whose parametric equations are $x=t^{2}+1, y=2 t$ and $x=2 s, y=\frac{2}{s}$ is given by
(a) $(1,-3)$
(b) $(2,2)$
(c) $(-2,4)$
(d) $(1,2)$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
02:37

Problem 105

Equation of the circumcircle of the triangle whose vertices are $(0,0),(1,0),(-1,-3)$ is
(a) $x^{2}+y^{2}-3 x+11 y=0$
(b) $3\left(x^{2}+y^{2}-x\right)+11 y=0$
(c) $3\left(x^{2}+y^{2}\right)+x-11 y=0$
(d) $3\left(x^{2}+y^{2}\right)-x+11 y=0$

Harshita Goel
Harshita Goel
Numerade Educator
02:35

Problem 106

Locus of the foot of the perpendiculars drawn from the centre of the circle $x^{2}+y^{2}+2 g x+2 f y+2 c=0$ to its chords which subtend a right angle at the centre is
(a) $(x+g)^{2}+(y+f)^{2}=f^{2}+g^{2}-c$
(b) $(x+g)^{2}+(y+f)^{2}=\frac{c}{2}$
(c) $(x+g)^{2}+(y+f)^{2}=2\left(f^{2}+g^{2}-c\right)$
(d) $(x+g)^{2}+(y+f)^{2}=\frac{f^{2}+g^{2}-c}{2}$

Harshita Goel
Harshita Goel
Numerade Educator
03:14

Problem 107

A line passing through $\mathrm{P}(3,5)$ intersects the circle $\mathrm{x}^{2}+\mathrm{y}^{2}=4$ at $\mathrm{A}$ and $\mathrm{B}$. A point $\mathrm{Q}$ is taken on $\mathrm{AB}$ such that $2 \mathrm{PQ}=$ $\mathrm{PA}+\mathrm{PB}$. Then, the locus of $\mathrm{Q}$ is
(a) $x^{2}+y^{2}-3 x+5 y=0$
(b) $x^{2}+y^{2}+3 x-5 y=0$
(c) $x^{2}+y^{2}+3 x+5 y=0$
(d) $x^{2}+y^{2}-3 x-5 y=0$

Harshita Goel
Harshita Goel
Numerade Educator
01:56

Problem 108

Circum circle of an equilateral triangle $\mathrm{ABC}$ is $\mathrm{x}^{2}+\mathrm{y}^{2}+2 \mathrm{gx}+2 \mathrm{fy}=0$. Then its incircle is
(a) $x^{2}+y^{2}+2 g x+2 f y=0$
(b) $4\left(x^{2}+y^{2}\right)+8 g x+8 f y+3 g^{2}+3 f^{2}=0$
(c) $x^{2}+y^{2}+8 g x+8 f y+3 g^{2}+3 f^{2}=0$
(d) $x^{2}+y^{2}+2 g x+2 f y+3 g^{2}+3 f^{2}=0$

Harshita Goel
Harshita Goel
Numerade Educator
02:06

Problem 109

The equation of the circle touching the lines given by $|x-2|+|y-3|=4$ is
(a) $x^{2}+y^{2}+4 x+6 y-5=0$
(b) $x^{2}+y^{2}-4 x-6 y+5=0$
(c) $x^{2}+y^{2}-4 x+6 y-5=0$
(d) $x^{2}+y^{2}-4 x-6 y-5=0$

Harshita Goel
Harshita Goel
Numerade Educator
01:50

Problem 110

The equation of the circle with latus rectum of $y^{2}+4 y+4 x+2=0$ as a diameter is
(a) $4\left(x^{2}+y^{2}+x+4 y\right)+1=0$
(b) $4\left(x^{2}+y^{2}+4 y\right)+1=0$
(c) $x^{2}+y^{2}+x+4 y+1=0$
(d) $x^{2}+y^{2}+3 x+4 y+1=0$

Harshita Goel
Harshita Goel
Numerade Educator
02:36

Problem 111

Each question contains Statement- 1 and Statement- 2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct.
(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True
Statement 1 A common tangent to the circles $x^{2}+y^{2}-6 x=0$ and $x^{2}+y^{2}-10 x+9=0$ is given by $2 y=x+3$
and
Statement 2
If two circles touch each other, their radical axis is the common tangent.

Harshita Goel
Harshita Goel
Numerade Educator
02:00

Problem 112

Each question contains Statement- 1 and Statement- 2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct.
(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True
Statement 1
The equation $x y+4 x+2 y+8=0$ represents a rectangular hyperbola
and
Statement 2
$a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0$ represents a rectangular hyperbola if $a+b=0$ and $a b c+2 f g h-a f^{2}-b g^{2}-c h^{2}$ $\neq 0$

Harshita Goel
Harshita Goel
Numerade Educator
03:28

Problem 113

Each question contains Statement- 1 and Statement- 2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct.
(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True
Statement 1 The foci of the ellipse $\frac{(x-2)^{2}}{25}+\frac{(y-3)^{2}}{49}=1$ are at $(2,3+2 \sqrt{6})$ and $(2,3-2 \sqrt{6})$
and
Statement 2 The foci of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ where $a>b$ are at $(a e, 0)$ and $(-a e, 0)$

Harshita Goel
Harshita Goel
Numerade Educator
04:24

Problem 114

The equation of the curve is
(a) $y^{2}=x, x \in[0, \infty)$
(b) $\mathrm{y}^{2}=2 \mathrm{x}, \mathrm{x} \in[0, \infty)$
(c) $y^{2}=4 x, x \in[0, \infty)$
(d) $x^{2}+y^{2}=1, x \in R$

Harshita Goel
Harshita Goel
Numerade Educator
05:41

Problem 115

Orthocentre of triangle PAS is
(a) $\mathrm{P}$
(b) A
(c) $\mathrm{S}$
(d) mid-point of PS

Harshita Goel
Harshita Goel
Numerade Educator
02:34

Problem 116

It is stated that a bisector of the angle between line PS and the line through P parallel to the $x$ -axis is
(i) the line PN
(ii) the line PT but not line PN
(iii) the lines PN and PT
(iv) the line PN but not line PT
Which of the following is true?
(a) (i) and (iv)
(b) (ii)
(c) (i) and (iii)
(d) None of these

Harshita Goel
Harshita Goel
Numerade Educator
04:29

Problem 117

If $\theta$ is the angle subtended at $\mathrm{P}(2,3)$ by the circle $\mathrm{x}^{2}+\mathrm{y}^{2}+2 \mathrm{x}-3 \mathrm{y}+1=0$, then
(a) $\cot \left(\frac{\theta}{2}\right)=2$
(b) $\tan \left(\frac{\theta}{2}\right)=\frac{1}{\sqrt{2}}$
(c) $\tan \theta=\frac{4}{3}$
(d) $\sec \theta=3$

Harshita Goel
Harshita Goel
Numerade Educator
04:09

Problem 118

The equation of a conic is $25 \mathrm{x}^{2}+16 \mathrm{y}^{2}=1600$. Then
(a) its latus rectum $=25$
(b) focal distances of the point $(4 \sqrt{3}, 5)$ are 7 and 13
(c) line $x+4 y+4 c=0$ is a tangent to the conic if $c^{2}=\frac{281}{4}$
(d) line $2 x+8 y+3=0$ is a tangent to the conic

Harshita Goel
Harshita Goel
Numerade Educator
02:49

Problem 119

The equation $\frac{x^{2}}{16-k}+\frac{y^{2}}{6-k}=1$ represents
(a) an ellipse if $\mathrm{k}<6$
(b) an ellipse if $\mathrm{k}>16$ or $\mathrm{k}<6$
(c) a hyperbola if $6<\mathrm{k}<16$
(d) if $\mathrm{k}=11$, the equation represents a rectangular hyperbola

Harshita Goel
Harshita Goel
Numerade Educator
09:45

Problem 120

Match the elements of Column I to elements of Column II. There can be single or multiple matches.
Column I
(a) The circles $x^{2}+y^{2}+6 x+k=0$ and $x^{2}+y^{2}+8 y-20=0$ are touching each other, if $\mathrm{k}$ equals
(b) Shortest distance from $(2,-7)$ to the circle $x^{2}+y^{2}-14 x-10 y-151=0$ is
(c) If the two circles $x^{2}+y^{2}+12 x+10 y+45=0$ and $x^{2}+y^{2}+2 x-14 y+a=0$ are touching each other externally, then the length of direct common tangent of the circles is
(d) The circles $x^{2}+y^{2}=4$ and $x^{2}+y^{2}-4 \lambda x+9=0$ have exactly two common tangents if $\lambda$ equals
Column II
(p) 8
(q) 2
(r) 12
(s) $-112$

Harshita Goel
Harshita Goel
Numerade Educator
04:13

Problem 121

(i) Find the equation of the circle touching the line $3 x+4 y=25$ at $(3,4)$ and having radius 5 units.
(ii) If there exists more than one circle find common tangents of these circles.

Harshita Goel
Harshita Goel
Numerade Educator
03:27

Problem 122

(i) The straight line $\frac{x}{p}+\frac{y}{q}=1$ is a chord of the circle $x^{2}+y^{2}=r^{2}$. Find the length of this chord.
(ii) Hence find the condition for the line $\frac{x}{p}+\frac{y}{q}=1$ to touch the circle $x^{2}+y^{2}=r^{2}$.

Harshita Goel
Harshita Goel
Numerade Educator
04:11

Problem 123

(i) From any point on the circle $x^{2}+y^{2}+2 g x+2 f y+c=0$, tangents are drawn to the circle $x^{2}+y^{2}+2 g x+2 f y+c \sin ^{2} \alpha$ $+\left(\mathrm{g}^{2}+\mathrm{f}^{2}\right) \cos ^{2} \alpha=0$ Find the angle between these tangents.
(ii) If the circle $x^{2}+y^{2}+2 g x+2 f y+c=0$ is the director circle of the circle $x^{2}+y^{2}+2 g x+2 f y+k=0$, find the value of $\mathrm{k}$.

Harshita Goel
Harshita Goel
Numerade Educator
03:30

Problem 124

(i) Find the equations of the common tangents of the circle $(x-3)^{2}+y^{2}=9$ and $y^{2}=4 x$.
(ii) Find the area bounded by these tangents.

Harshita Goel
Harshita Goel
Numerade Educator
05:32

Problem 125

(i) If the normal to the parabola $y^{2}=4 a x$ at the point $\left(a t^{2}, 2 a t\right)$ meets the parabola again at $\left(a t_{1}^{2}, 2 a t_{2}\right)$ then prove that $t_{1}=-t-\frac{2}{t}$
(ii) Find the minimum length of such chords.
(iii) Find the equation(s) of these shortest normal chord(s).

Harshita Goel
Harshita Goel
Numerade Educator
04:16

Problem 126

(i) Find the area of the triangle inscribed in the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, eccentric angles of whose vertices are $\alpha, \beta$ and $\gamma$.
(ii) Find the maximum area of a triangle inscribed in the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$.

Harshita Goel
Harshita Goel
Numerade Educator
04:05

Problem 127

If a point $\mathrm{P}$ on the ellipse $\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}+\frac{\mathrm{y}^{2}}{\mathrm{~b}^{2}}=1$ is joined to $\mathrm{S}_{1}$ and $\mathrm{S}_{2}$ (foci of the ellipse) then,
(i) show that the incentre of the triangle $\mathrm{PS}_{1} \mathrm{~S}_{2}$ lies on another ellipse.
(ii) find the eccentricity of this ellipse in terms of the eccentricity of the given ellipse.

Harshita Goel
Harshita Goel
Numerade Educator
02:22

Problem 128

(i) Find the locus of a point of the form $\left(e^{t}+e^{-t}, e^{t}-e^{-t}\right)$
(ii) Find the eccentricity of the locus
(iii) Find the slope of the tangent drawn to the curve at $\mathrm{t}=1$.

Harshita Goel
Harshita Goel
Numerade Educator
04:01

Problem 129

If the circle $x^{2}+y^{2}+2 g x+2 f y+c=0$ cuts the curve $x=p t, y=\frac{p}{t}(t$ being a parameter) at $A, B, C$ and $D$ with parameters $\mathrm{t}_{1}, \mathrm{t}_{2}, \mathrm{t}_{3}, \mathrm{t}_{4}$, prove that
(i) $\mathrm{t}_{1} \mathrm{t}_{2} \mathrm{t}_{3} \mathrm{t}_{4}=1$
(ii) $\frac{1}{t_{1}}+\frac{1}{t_{2}}+\frac{1}{t_{3}}+\frac{1}{t_{4}}=\frac{-2 f}{p}$

Harshita Goel
Harshita Goel
Numerade Educator
03:44

Problem 130

The tangents to the rectangular hyperbola $x y=c^{2}$ and the parabola $y^{2}=4 a x$ at their point of intersection are inclined at angles $\alpha$ and $\beta$ respectively to the $x$ -axis. Show that $\tan \alpha+2 \tan \beta=0$.

Harshita Goel
Harshita Goel
Numerade Educator
01:17

Problem 131

If one end of a diameter of the circle $x^{2}+y^{2}-10 x+6=0$ is at $(1,2)$, then the other end is at
(a) $(9,-2)$
(b) $(2,-9)$
(c) $(5,0)$
(d) $(9,0)$

Goutam Chand
Goutam Chand
Numerade Educator
01:47

Problem 132

The angle between the asymptotes of $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ is equal to
(a) $\tan ^{-1} \mathrm{a}$
(b) $\tan ^{-1} \mathrm{~b}$
(c) $\tan ^{-1} \frac{\mathrm{b}}{\mathrm{a}}$
(d) $2 \tan ^{-1}\left(\frac{\mathrm{b}}{\mathrm{a}}\right)$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:31

Problem 133

Slope of the tangent at the point $\left(\frac{a}{2}, \frac{\sqrt{3} a}{2}\right)$ on the circle $x^{2}+y^{2}=a^{2}$ is
(a) 1
(b) $\frac{\sqrt{3}}{2}$
(c) $\frac{-1}{\sqrt{3}}$
(d) $-\sqrt{3}$

Goutam Chand
Goutam Chand
Numerade Educator
03:37

Problem 134

Two vertices of an equilateral triangle are at $(-1,0)$ and at $(1,0)$ and its third vertex lies above the $\mathrm{x}$ -axis. The equation of its circumcircle is
(a) $(x-0)^{2}+y^{2}=\frac{2}{\sqrt{3}}$
(b) $(x-0)^{2}+\left(y-\frac{1}{\sqrt{3}}\right)^{2}=\left(\frac{2}{\sqrt{3}}\right)^{2}$
(c) $\left(x-\frac{1}{\sqrt{3}}\right)^{2}+(y-0)^{2}=1$
(d) $\left(x-\frac{1}{\sqrt{3}}\right)^{2}+\left(y+\frac{1}{\sqrt{3}}\right)^{2}=1$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
02:05

Problem 135

The circles $x^{2}+y^{2}-x-y=0$ and $x^{2}+y^{2}-x+y=0$ intersect at an angle
(a) $\frac{\pi}{2}$
(b) $\frac{\pi}{3}$
(c) $\frac{\pi}{6}$
(d) $\frac{\pi}{4}$

Goutam Chand
Goutam Chand
Numerade Educator
02:31

Problem 136

Locus of point of intersection of perpendicular tangents to the circle $x^{2}+y^{2}-4 x-6 y-12=0$
(a) $x^{2}+y^{2}-4 x-6 y+37=0$
(b) $x^{2}+y^{2}-4 x-6 y-37=0$
(c) $x^{2}+y^{2}+4 x+6 y-37=0$
(d) $x^{2}+y^{2}-4 x-6 y-25=0$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:59

Problem 137

$\mathrm{P}$ and $\mathrm{Q}$ are two points on the line through $(2,4)$ and having slope $\mathrm{m}$. If the line segment $\mathrm{AB}$ joining $(0,0)$ and $(6,0)$ subtends right angles at $\mathrm{P}$ and $\mathrm{Q}$ then
(a) $\mathrm{m} \notin\left[\frac{2-3 \sqrt{2}}{4}, \frac{2+3 \sqrt{2}}{4}\right]$
(b) $\frac{2-3 \sqrt{2}}{4}<\mathrm{m}<\frac{2+3 \sqrt{2}}{4}$
(c) $\frac{2-3 \sqrt{2}}{4} \leq \mathrm{m} \leq \frac{2+3 \sqrt{2}}{4}$
(d) $\mathrm{m} \in \mathrm{R}$

Harshita Goel
Harshita Goel
Numerade Educator
01:56

Problem 138

The triangle formed by the tangent to the parabola $y=x^{2}$ at the point $x=x_{0}\left(x_{0} \in[1,2]\right)$ the y-axis and the straight line $\mathrm{y}=\mathrm{x}_{0}{ }^{2}$ has the greatest area if $\mathrm{x}_{0}=$
(a) 1
(b) $\frac{4}{3}$
(c) $\frac{3}{2}$
(d) 2

Harshita Goel
Harshita Goel
Numerade Educator
05:56

Problem 139

If a normal to $\mathrm{y}^{2}=4 \mathrm{ax}$ makes an angle $\theta$ with the positive direction of $\mathrm{x}$ -axis, perpendicular distance of it from a tangent parallel to it is
(a) a $\operatorname{cosec}^{2} \theta \sec \theta$
(b) a $\operatorname{cosec} \theta \sec \theta$
(c) a $\operatorname{cosec} \theta \sec ^{2} \theta$
(d) a $\operatorname{cosec} \theta \sec ^{3} \theta$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
05:03

Problem 140

If a normal chord of $\mathrm{y}^{2}=4 \mathrm{x}$ subtends a right angle at the vertex, then the extremities of the chord are
(a) $(3, \pm 2 \sqrt{2}),(2, \pm 4 \sqrt{2})$
(b) $(2, \pm 2 \sqrt{2}),(8, \mp 4 \sqrt{2})$
(c) $(3,4 \sqrt{2}),(2, \sqrt{2})$
(d) $(3, \pm 2 \sqrt{2}),(8, \mp 4 \sqrt{2})$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
05:57

Problem 141

The tangents at $\mathrm{P}, \mathrm{Q}$ on $\mathrm{y}^{2}=4 \mathrm{ax}$ meet at $\mathrm{T}$, and $\mathrm{S}$ is the focus. If $\mathrm{SP}, \mathrm{ST}, \mathrm{SQ}$ are $\alpha, \beta, \gamma$ respectively, then the roots of $\alpha \mathrm{x}^{2}$ $+2 \beta x+\gamma=0$ are
(a) imaginary
(b) real and distinct integers
(c) real and equal
(d) real and irrational

Saurabh Chandra
Saurabh Chandra
Numerade Educator
03:23

Problem 142

Two vertices $\mathrm{A}$ and $\mathrm{B}$ of a triangle $\mathrm{ABC}$ are $(-3,0)$ and $(3,0) .$ The vertex $\mathrm{C}$ is varying in such a way that $4 \tan \frac{\mathrm{A}}{2} \tan \frac{\mathrm{B}}{2}=1$. The locus of $\mathrm{C}$ is
(a) $\frac{x^{2}}{16}+\frac{y^{2}}{25}=1$
(b) $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$
(c) $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$
(d) $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$

Harshita Goel
Harshita Goel
Numerade Educator
05:32

Problem 143

The product of the perpendiculars drawn from the foci on any tangent to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1,(a>b)$ is equal to
(a) $a^{2}$
(b) ab
(c) $\mathrm{a}^{2} \mathrm{~b}^{2}$
(d) $b^{2}$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
02:23

Problem 144

$\mathrm{P}(6,2 \sqrt{3})$ is a point on $\frac{\mathrm{x}^{2}}{9}-\frac{\mathrm{y}^{2}}{4}=1 .$ If $\mathrm{Q}$ is the projection of $\mathrm{P}$ on transverse axis, the point of contact of the tangent drawn from $\mathrm{Q}$ to the auxiliary circle of given hyperbola is
(a) $\left(\pm \frac{3}{2}, \frac{3 \sqrt{3}}{2}\right)$
(b) $\left(\pm \frac{3}{2}, \pm \frac{3 \sqrt{3}}{2}\right)$
(c) $\left(\frac{3}{2}, \pm \frac{3 \sqrt{3}}{2}\right)$
(d) $\left(-\frac{3}{2}, \frac{3 \sqrt{3}}{2}\right)$

Harshita Goel
Harshita Goel
Numerade Educator
04:24

Problem 145

Length of the common chord of the circles $(x-a)^{2}+y^{2}=a^{2}$ and $x^{2}+(y-b)^{2}=b^{2}$
(a) $\frac{2 a b}{a+b}$
(b) $\frac{a b}{a+b}$
(c) $\frac{2 a b}{\sqrt{a^{2}+b^{2}}}$
(d) $\frac{a b}{\sqrt{a^{2}+b^{2}}}$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
05:12

Problem 146

The distances of the origin, from the centres of three circles $x^{2}+y^{2}-2 \lambda x=c^{2}$ where $c$ is constant and $\lambda$ a variable parameter are in geometrical progression. Then the lengths of tangents drawn to them from any point on the circle $\mathrm{x}^{2}+\mathrm{y}^{2}=\mathrm{c}^{2}$ are in
(a) A.P
(b) G.P.
(c) H.P.
(d) not in any progression.

Harshita Goel
Harshita Goel
Numerade Educator
02:42

Problem 147

Tangents are drawn to the circles $x^{2}+y^{2}=a^{2}$ and $x^{2}+y^{2}=b^{2}$ at right angles to one another. The locus of their point of intersection is
(a) $x+y=a$
(b) $x^{2}+y^{2}=a^{2}+b^{2}$
(c) $a x+b y=1$
(d) $x^{2}+y^{2}+a b=0$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:46

Problem 148

The coordinates of any point on the circle through the points $\mathrm{A}(2,2), \mathrm{B}(5,3)$ and $\mathrm{C}(3,-1)$ can be written in the form $(4+\sqrt{5} \cos \theta, 1+\sqrt{5} \sin \theta) .$ Then, the coordinates of the point $\mathrm{P}$ on $\mathrm{BC}$ such that $\mathrm{AP}$ is perpendicular to $\mathrm{BC}$ are
(a) $(-1,4)$
(b) $(4,1)$
(c) $(1,4)$
(d) $(2,3)$

Harshita Goel
Harshita Goel
Numerade Educator
03:22

Problem 149

The normals at two points $\mathrm{P}$ and $\mathrm{Q}$ on the parabola $\mathrm{y}^{2}=4 \mathrm{ax}$ meet at a point $\mathrm{R}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)$ on the parabola then $\mathrm{PQ}^{2}$ is
(a) $\left(x_{1}-8 a\right)^{2}$
(b) $\left(x_{1}-8 a\right)\left(x_{1}+4 a\right)$
(c) $\left(x_{1}-4 a\right)\left(x_{1}+8 a\right)$
(d) $x_{1}^{2}-16 a^{2}$

Harshita Goel
Harshita Goel
Numerade Educator
04:45

Problem 150

The equation of a tangent to the ellipse $2 \mathrm{x}^{2}+7 \mathrm{y}^{2}=14$, drawn from the point $(5,2)$ is
(a) $x-9 y+13=0$
(b) $x-y-3=0$
(c) $x+y-3=0$
(d) both (a) and
(b)

Saurabh Chandra
Saurabh Chandra
Numerade Educator
03:18

Problem 151

$\mathrm{P}$ is a point on the ellipse $\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}+\frac{\mathrm{y}^{2}}{\mathrm{~b}^{2}}=1$ and $\mathrm{Q}$ is the corresponding point on the auxiliary circle. If the normals at $\mathrm{P}$ and Q meet in $\mathrm{R}$, then the locus of $\mathrm{R}$ is the circle $\mathrm{x}^{2}+\mathrm{y}^{2}$ is
(a) $\mathrm{a}^{2}+\mathrm{b}^{2}$
(b) $(a+b)^{2}$
(c) $\mathrm{a}^{2} \mathrm{~b}^{2}$
(d) ab

Harshita Goel
Harshita Goel
Numerade Educator
03:39

Problem 152

If the tangent at a point $\theta$ on an ellipse meets the auxiliary circle of the ellipse at two points, which subtend a right angle at the centre, the eccentricity of the ellipse is
(a) $1+\sin ^{2} \theta$
(b) $\sin ^{2} \theta$
(c) $\left(1+\sin ^{2} \theta\right)^{-1 / 2}$
(d) $(1+\sin \theta)^{1 / 2}$

Harshita Goel
Harshita Goel
Numerade Educator
05:05

Problem 153

If the chord joining the variable points ' $\alpha$ ' and ' $\beta$ ' on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ subtends a right angle at the point $(a, 0)$ then $\tan \frac{\alpha}{2} \tan \frac{\beta}{2}$ is
(a) $\frac{-b}{a}$
(b) $\frac{\mathrm{b}^{2}}{\mathrm{a}^{2}}$
(c) $\frac{-a^{2}}{b^{2}}$
(d) $\frac{-b^{2}}{a^{2}}$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
02:39

Problem 154

The locus of the mid points of chords of the hyperbola $x y=c^{2}$ which pass through the fixed point $(2 p, 2 q)$ is the hyperbola $(x-p)(y-q)=$
(a) $\mathrm{p}+\mathrm{q}$
(b) $2 \mathrm{pq}$
(c) $\mathrm{pq}$
(d) $\mathrm{p}^{2}+\mathrm{q}^{2}$

Harshita Goel
Harshita Goel
Numerade Educator
02:36

Problem 155

The equation of a circle touching the lines $x-y=3, x+y=3$ and passing through $(5, \sqrt{2})$ is
(a) $x^{2}+y^{2}=27$
(b) $(x-10)^{2}+y^{2}=27$
(c) $(x-9)^{2}+y^{2}=18$
(d) $(x-3)^{2}+y^{2}=6$

Harshita Goel
Harshita Goel
Numerade Educator
03:54

Problem 156

The conditions that the two circles passing through $(0, a)$ and $(0,-a)$ and touching the line $y=m x+c$ are orthogonal, is
(a) $c^{2}=a^{2} m^{2}$
(b) $c^{2}=a^{2}\left(1+m^{2}\right)$
(c) $c^{2}=2 \mathrm{am}^{2}$
(d) $c^{2}=a^{2}\left(2+m^{2}\right)$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
05:22

Problem 157

The line $2 \mathrm{x}-\mathrm{y}+4=0$ is a diameter of the circle which circumscribes a rectangle $\mathrm{ABCD}$. If the co-ordinates of $\mathrm{A}$ and $B$ are $A(4,6), B(1,9)$, then the area of rectangle $A B C D$ is
(a) 18
(b) 8
(c) 16
(d) 12

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:38

Problem 158

If two circles $S=x^{2}+y^{2}+2 a x+2 b y+c=0$, and $S^{\prime}=x^{2}+y^{2}+2 b x+2 a y+c=0$ are intersecting each other, then
(a) $(a+b)^{2}>2 c$
(b) $(a+b)^{2}<2 c$
(c) $(a+b)^{2}<2 b$
(d) $b+c<c$

P Krishnamurthy
P Krishnamurthy
Numerade Educator
02:18

Problem 159

If the normals at the points $\mathrm{P}, \mathrm{Q}, \mathrm{R}$ on the parabola $\mathrm{y}^{2}=4 \mathrm{ax}$ intersect at the point $(\alpha, \beta)$, then the orthocentre of triangle $\mathrm{PQR}$ is
(a) $\left(\frac{2}{3} \mathrm{a}, 0\right)$
(b) $\left(\alpha-3 \mathrm{a}, \frac{-\beta}{2}\right)$
(c) $\left(\alpha-6 \mathrm{a}, \frac{-\beta}{2}\right)$
(d) $\left(\frac{\alpha}{2}, \frac{\beta}{2}\right)$

Mahipal Kumawat
Mahipal Kumawat
Numerade Educator
01:49

Problem 160

If the circles $x^{2}+y^{2}=\lambda^{2}$ and $x^{2}+y^{2}-6 x-8 y+9=0$ touch externally, then $\lambda$ is
(a) $-2$
(b) 3
(c) 2
(d) 1

Saurabh Chandra
Saurabh Chandra
Numerade Educator
02:57

Problem 161

The equation of the tangent to the circle $x^{2}+y^{2}=16$ such that the corresponding normal passes through $(2,4)$ is
(a) $2 x-y \pm 4 \sqrt{5}=0$
(b) $x+2 y \pm 4 \sqrt{5}=0$
(c) $\quad x-2 y \pm 4 \sqrt{5}=0$
(d) $2 x+y \pm 4 \sqrt{5}=0$

Goutam Chand
Goutam Chand
Numerade Educator
01:07

Problem 162

The equation of the parabola whose vertex is at $(2,1)$ and whose focus is at $(1,-1)$ is
(a) $4 x^{2}-4 x y+y^{2}-32 x-34 y+89=0$
(b) $4 x^{2}-4 x y+y^{2}+8 x+46 y-71=0$
(c) $4 x^{2}+4 x y+y^{2}-32 x-34 y+89=0$
(d) $4 x^{2}+4 x y+y^{2}+8 x+46 y-71=0$

AG
Ankit Gupta
Numerade Educator
01:30

Problem 163

The two curves whose equations are given by $7 x^{2}+16 y^{2}=112$ and $225 x^{2}-400 y^{2}=1296$
(a) have their latus rectum equal
(b) have the same foci
(c) have the same vertices
(d) have the same directrices

AG
Ankit Gupta
Numerade Educator
05:03

Problem 164

Let $\mathrm{P}$ be the mid point of a chord of $\mathrm{x}^{2}+\mathrm{y}^{2}=8$ such that this chord cuts the curve $\mathrm{x}^{2}-2 \mathrm{x}-2 \mathrm{y}=0$ at $\mathrm{A}$ and $\mathrm{B}$. Also, AB subtends a right angle at the origin. Then $\mathrm{P}$ is
(a) $(1, \sqrt{2})$
(b) $(2,0)$
(c) $(0,-2)$
(d) $(2,2)$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
01:34

Problem 165

The equation of a common tangent to the parabola $\mathrm{y}^{2}=8 \mathrm{x}$ and the hyperbola $81 \mathrm{x}^{2}-10 \mathrm{y}^{2}=40$ is
(a) $3 y=9 x+1$
(b) $3 y=-9 x-1$
(c) $3 y=9 x+2$
(d) $3 x+3 y=2$

Patha Sharma
Patha Sharma
Numerade Educator
02:30

Problem 166

If the line $1 x+m y+n=0$ is a normal to the curve $x y=c^{2}$ (where $c$ is a constant), then
(a) $\operatorname{lm}<0$
(b) $1+\mathrm{m}>0, \mathrm{n}>0$
(c) $1 \leq 0 ; \mathrm{m} \leq 0$
(d) $1>0, \mathrm{~m}>0$

Saurabh Chandra
Saurabh Chandra
Numerade Educator
04:23

Problem 167

$\mathrm{AB}$ is a chord of the circle $\mathrm{x}^{2}+\mathrm{y}^{2}-4 \mathrm{x}-2 \mathrm{y}+1=0$ subtending a right angle at the centre. Then the locus of the centroid of the triangle PAB, where $\mathrm{P}$ moves on the circle is
(a) a parabola
(b) an ellipse
(c) a hyperbola
(d) a circle

Saurabh Chandra
Saurabh Chandra
Numerade Educator
06:52

Problem 168

Line $L$ is the tangent to the circle $x^{2}+y^{2}=1$ at $(0,1)$. A ray of light incident at the point $(3,1)$ gets reflected in $L$. If the reflected ray touches the circle, then the equation of the incident ray is
(a) $3 x-4 y=5$
(b) $3 x+4 y=13$
(c) $4 x-3 y=9$
(d) $4 x+3 y=15$

Anurag Kumar
Anurag Kumar
Numerade Educator
03:42

Problem 169

A man is walking along the boundary of the circular path, $x^{2}+y^{2}-2 x-2 y=0$. If the speed of the man is $\frac{\pi}{3} \mathrm{~m} / \mathrm{sec}$
and he starts from $\mathrm{P}(2,2)$ and moves in the anti-clockwise direction, then the position of the man after $\frac{5}{\sqrt{2}}$ sec is
(a) $\left(\frac{1-\sqrt{3}}{2}, \frac{3-\sqrt{3}}{2}\right)$
(b) $\left(\frac{3-\sqrt{3}}{2}, \frac{3+\sqrt{3}}{2}\right)$
(c) $\left(\frac{-3+\sqrt{3}}{2}, \frac{-3-\sqrt{3}}{2}\right)$
(d) $\left(\frac{-1-\sqrt{3}}{2}, \frac{-3+\sqrt{3}}{2}\right)$

Harshita Goel
Harshita Goel
Numerade Educator
01:28

Problem 170

The points of contact of the tangents drawn from $\mathrm{P}(-2,2)$ to the parabola $\mathrm{y}^{2}=16 \mathrm{x}$ are $\mathrm{A}$ and $\mathrm{B}$. Then $\mathrm{AB}$ is
(a) $4 \sqrt{17}$
(b) $\sqrt{5}$
(c) $2 \sqrt{17}$
(d) $3 \sqrt{17}$

Hast Aggarwal
Hast Aggarwal
Numerade Educator
03:51

Problem 171

Each question contains Statement- 1 and Statement- 2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct.
(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True
Statement 1 The line $x+1=0$ is a common tangent to the circle $x^{2}+y^{2}-2 x-3=0$ and the parabola $y^{2}+8 x-4 y+12=0$
and
Statement 2
The line $\mathrm{y}=\mathrm{m} \mathrm{x}+\mathrm{c}$ is a tangent to the parabola $\mathrm{y}^{2}+4 \mathrm{ax}=0$ if $\mathrm{c}=\frac{\mathrm{a}}{\mathrm{m}}$

Harshita Goel
Harshita Goel
Numerade Educator
02:54

Problem 172

Each question contains Statement- 1 and Statement- 2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct.
(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True
Statement 1 Circles $C_{1}: x^{2}+y^{2}-2 x+4 y-31=0$ and
$C_{2}: x^{2}+y^{2}-8 x-4 y-149=0$
are such that $C_{1}$ is inside $C_{2}$
and
Statement 2
If two circles are such that distance between their centres is greater than the sum of their radii, then they do not intersect.

Harshita Goel
Harshita Goel
Numerade Educator
02:06

Problem 173

Each question contains Statement- 1 and Statement- 2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct.
(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True
Statement 1 $4 x-3 y-5=0$ is a chord of the circle $x^{2}+y^{2}-6 x-8 y+5=0$ through $(2,1)$
and
Statement 2 Perpendicular distance from $(4,-3)$ to $4 x-3 y-5=0$ is 4 .

Harshita Goel
Harshita Goel
Numerade Educator
02:19

Problem 174

Each question contains Statement- 1 and Statement- 2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct.
(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True
Statement 1 Given two circles $S_{1}=0$ and $S_{2}=0$ having same coefficients in $x^{2}$ and $y^{2}$, then $S_{1}-S_{2}=0$ is the equation of a common tangent of the circles.
and
Statement 2
$4 \mathrm{x}+3 \mathrm{y}-8=0$ is a common tangent of the circles $x^{2}+y^{2}+2 x+2 y-7=0$ and $x^{2}+y^{2}-6 x-4 y+9=0$

Harshita Goel
Harshita Goel
Numerade Educator
01:45

Problem 175

Each question contains Statement- 1 and Statement- 2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct.
(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True
Statement 1 Distance between the foci of any ellipse of the family $x^{2} \cos ^{2} \theta+y^{2} \cot ^{2} \theta=1$ is a constant.
and
Statement 2
Sum of focal distances of point on an ellipse of the family $x^{2} \cos ^{2} \theta+y^{2} \cot ^{2} \theta=1$ is not a constant.

Harshita Goel
Harshita Goel
Numerade Educator
02:33

Problem 176

Each question contains Statement- 1 and Statement- 2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct.
(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True
Statement 1 The centre of the circle $x^{2}+y^{2}-6 x+2 y+1=0$ lies on the pair of lines represented by the equation $x^{2}+x y-2 y^{2}+$ $3 \mathrm{y}-1=0$, so that the lines are diameters of the circle.
and
Statement $\underline{2}$
If $(\alpha, \beta)$ be a point on any of the lines represented by $\mathrm{f}(\mathrm{x}, \mathrm{y})=0$ where, $f(x, y)=a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c$, then $f(\alpha, \beta)=0$

Harshita Goel
Harshita Goel
Numerade Educator
01:03

Problem 177

Each question contains Statement- 1 and Statement- 2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct.
(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True
Statement 1
The eccentricity of an ellipse is the ratio of the distance between the foci to the length of its major axis.
and
Statement 2
The eccentricity of an ellipse is independent of the length of minor axis.

Harshita Goel
Harshita Goel
Numerade Educator
01:24

Problem 178

Each question contains Statement- 1 and Statement- 2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct.
(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True
Statement 1
It is not possible to draw on a plane, a hyperbola and an ellipse having same foci.
and
Statement 2 $a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0$ represents a conic with eccentricity e, then for a hyperbola $e>1$ and for an ellipse $\mathrm{e}<1 .$

Harshita Goel
Harshita Goel
Numerade Educator
03:08

Problem 179

Each question contains Statement- 1 and Statement- 2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct.
(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True
Statement 1 Line perpendicular to $x$ axis through $P(3,1)$ on the ellipse $x^{2}+3 y^{2}=12$ cuts the circle $x^{2}+y^{2}=12$ at $Q$. Then tan $\angle \mathrm{POQ}$ is $\frac{3-\sqrt{3}}{3 \sqrt{3}+1}$
and
Statement 2
Eccentric angle $\theta$ of any point on an ellipse is the angle made by the ray joining the centre to the point on the auxiliary circle with same abscissa with positive direction of $x$ axis.

Harshita Goel
Harshita Goel
Numerade Educator
01:55

Problem 180

Each question contains Statement- 1 and Statement- 2 and has the following choices (a), (b), (c) and (d), out of which ONLY ONE is correct.
(a) Statement- 1 is True, Statement- 2 is True; Statement- 2 is a correct explanation for Statement-1
(b) Statement- 1 is True, Statement- 2 is True; Statement- 2 is NOT a correct explanation for Statement-1
(c) Statement- 1 is True, Statement- 2 is False
(d) Statement- 1 is False, Statement- 2 is True
Statement 1
The line $3 x-2 y=0$ will not meet the curve $9 x^{2}-4 y^{2}=36$.
and
Statement 2
$\mathrm{y}=\frac{\mathrm{b}}{\mathrm{a}} \mathrm{x}$ is an asymptote of the hyperbola $\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}-\frac{\mathrm{y}^{2}}{\mathrm{~b}^{2}}=1$.

Harshita Goel
Harshita Goel
Numerade Educator
02:58

Problem 181

The centre of the ellipse $17 x^{2}-12 x y+8 y^{2}+46 x-28 y+17=0$ is at
(a) $(1,1)$
(b) $(-1,1)$
(c) $(-1,-1)$
(d) $(1,-1)$

Saurabh Kumar Gupta
Saurabh Kumar Gupta
Numerade Educator
16:35

Problem 182

If the axes are rotated through an angle $\frac{\pi}{4}$ in the counter clockwise sense without shifting the origin, the equation $x^{2}-3 x y+y^{2}+10 x-10 y+21=0$ reduces to
(a) $\quad X^{2}-5 Y^{2}-20 \sqrt{2} X+20 \sqrt{2} Y-42=0$
(b) $X^{2}+5 Y^{2}-20 \sqrt{2} X+20 \sqrt{2} Y-42=0$
(c) $\quad X^{2}-5 Y^{2}+10 \sqrt{2} X-10 \sqrt{2} Y-21=0$
(d) None of the above

Chris Trentman
Chris Trentman
Numerade Educator
02:05

Problem 183

The eccentricity of the ellipse represented by the equation $4(x-2 y+1)^{2}+9(2 x+y+2)^{2}=25$ is
(a) $\frac{5}{9}$
(b) $\frac{\sqrt{5}}{3}$
(c) $\frac{3}{2 \sqrt{5}}$
(d) None of these

Goutam Chand
Goutam Chand
Numerade Educator
03:29

Problem 184

Let the second-degree equation $a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0$ represent an ellipse. Which of the following is are invariant under the shifting of origin and the rotation of co-ordinate axes?
(i) $\mathrm{h}^{2}-\mathrm{ab}$
(ii) $a+b$
(iii) eccentricity
(iv) coordinates of foci
(a) (i) and (ii)
(b) (iii) and (iv)
(c) (i), (ii) and (iii)
(d) all the four.

Charles Carter
Charles Carter
Numerade Educator
01:26

Problem 185

All the points on the ellipse $\frac{\mathrm{x}^{2}}{2}+\frac{\mathrm{y}^{2}}{1}=1$ are rotated about origin through an angle $\frac{\pi}{4}$ in counter clockwise sense. The equation of ellipse changes to
(a) $3 \mathrm{X}^{2}+3 \mathrm{Y}^{2}+2 \mathrm{XY}=4$
(b) $3 \mathrm{X}^{2}+3 \mathrm{Y}^{2}+\mathrm{X} Y=2$
(c) $3 \mathrm{X}^{2}+3 \mathrm{Y}^{2}-2 \mathrm{XY}=4$
(d) $\frac{X^{2}}{\sqrt{2}}+\frac{Y^{2}}{\sqrt{2}}-X Y=1$

Eva R
Eva R
Numerade Educator
01:26

Problem 186

All the points on the ellipse of question no.10, after rotation are then shifted two units along positive $\mathrm{x}$ -direction and two units along positive y-direction. The equation then becomes
(a) $3 \mathrm{X}^{2}+3 \mathrm{Y}^{2}+2 \mathrm{X}+2 \mathrm{Y}+2 \mathrm{XY}=4$
(b) $3 \mathrm{X}^{2}+3 \mathrm{Y}^{2}+2 \mathrm{X}+2 \mathrm{Y}+\mathrm{XY}=2$
(c) $\frac{\mathrm{X}^{2}}{\sqrt{2}}+\frac{\mathrm{Y}^{2}}{\sqrt{2}}-2 \mathrm{X}-2 \mathrm{Y}-\mathrm{XY}=1$
(d) $3 \mathrm{X}^{2}+3 \mathrm{Y}^{2}-8 \mathrm{X}-8 \mathrm{Y}-2 \mathrm{XY}+12=0$

Eva R
Eva R
Numerade Educator
03:08

Problem 187

Eccentricity of the ellipse is
(a) $\frac{\sqrt{6}}{5}$
(b) $\frac{2 \sqrt{6}}{5}$
(c) $\frac{3}{5}$
(d) $\frac{\sqrt{6}}{10}$

Saurabh Kumar Gupta
Saurabh Kumar Gupta
Numerade Educator
01:48

Problem 188

Equation of the minor axis of the ellipse is
(a) $3 x+4 y-1=0$
(b) $4 x+3 y-6=0$
(c) $6 x+8 y-19=0$
(d) $6 x-8 y+19=0$

Saurabh Kumar Gupta
Saurabh Kumar Gupta
Numerade Educator
01:46

Problem 189

If $C$ denotes the circle on $\mathrm{AB}$ as diameter, length of tangent from the point $(2,6)$ to $\mathrm{C}$ is given by
(a) $2 \sqrt{3}$
(b) $4 \sqrt{3}$
(c) 6
(d) point is inside $\mathrm{C}$

Saurabh Kumar Gupta
Saurabh Kumar Gupta
Numerade Educator
03:39

Problem 190

The tangent at $\mathrm{P}(5 \cos \theta, 3 \sin \theta)$ on the ellipse $\frac{\mathrm{x}^{2}}{25}+\frac{\mathrm{y}^{2}}{9}=1$ meets the auxiliary circle $\mathrm{x}^{2}+\mathrm{y}^{2}=25$ at two points $\mathrm{A}$ and $\mathrm{B}$.
If AB subtends $90^{\circ}$ at the centre of the ellipse, then
(a) $\sin ^{2} \theta=\frac{9}{16}$
(b) $\sin ^{2} \theta=\frac{16}{25}$
(c) $\mathrm{e}^{2}=\left(1+\sin ^{2} \theta\right)^{-1}$
(d) $\mathrm{e}^{2}=\left(1+\cos ^{2} \theta\right)^{-1}$

Harshita Goel
Harshita Goel
Numerade Educator
01:28

Problem 191

A circle of radius a, centre $(0,0)$ touches the directrix of $y^{2}=4 a x$ at $P$. Tangents are drawn from $P$ to the parabola touches it at $\mathrm{Q}$ and $\mathrm{R}$. Then
(a) othocenter of $\triangle \mathrm{PQR}$ is $(\mathrm{a}, 0)$
(b) othocenter of $\Delta \mathrm{PQR}$ is $(-\mathrm{a}, 0)$
(c) circumcenter of $\Delta \mathrm{PQR}$ is $(\mathrm{a}, 0)$
(d) circumcenter of $\Delta \mathrm{PQR}$ is $(-\mathrm{a}, 0)$

Saurabh Kumar Gupta
Saurabh Kumar Gupta
Numerade Educator
01:24

Problem 192

The line $\mathrm{x}-\mathrm{y}+2 \mathrm{c}=0$ intersects the director circle of $\mathrm{x}^{2}+\mathrm{y}^{2}=\mathrm{a}^{2},(\mathrm{a}>0)$ in two real points if
(a) $c+a<0$
(b) $c+a>0$
(c) $c-a<0$
(d) $c-a>0$

Saurabh Kumar Gupta
Saurabh Kumar Gupta
Numerade Educator
00:54

Problem 193

A line which touches both $\mathrm{y}^{2}-8 \mathrm{x}=0$ and $\mathrm{x}^{2}+\mathrm{y}^{2}+8 \mathrm{x}+14=0$ is
(a) $x+y+2=0$
(b) $x+y-2=0$
(c) $x-y+2=0$
(d) $x-y-2=0$

Saurabh Kumar Gupta
Saurabh Kumar Gupta
Numerade Educator
02:37

Problem 194

Given $x^{2}+y^{2}=8 x$ represents the circle $C$ and $y^{2}=4 x$ represents the parabola $P$. Then
(a) $(2,3)$ is point is region bounded by $\mathrm{C}$ and $\mathrm{P}$
(b) $(2,-3)$ is a point inside of both $\mathrm{C}$ and $\mathrm{P}$
(c) $(2,4)$ is a point outside of both $\mathrm{C}$ and $\mathrm{P}$
(d) $(2,-4)$ is a point is a region bounded by $C$ and $P$

Aman Gupta
Aman Gupta
Numerade Educator
00:32

Problem 195

An ellipse is orthogonal to the hyperbola $x^{2}-y^{2}=2 .$ The eccentricity of the ellipse is reciprocal of that of the hyperbola. Then
(a) equation of the ellipse is $x^{2}+2 y^{2}=8$
(b) focus of the ellipse is at $(-4 \sqrt{2}, 0)$
(c) directrix of the ellipse is $x+4 \sqrt{2}=0$
(d) directrix circle of the ellipse $x^{2}+y^{2}=12$

Himanshu Kushwaha
Himanshu Kushwaha
Numerade Educator
01:18

Problem 196

Auxiliary circle of the ellipse $\frac{\mathrm{x}^{2}}{\mathrm{a}^{2}}+\frac{\mathrm{y}^{2}}{\mathrm{~b}^{2}}=1(\mathrm{a}>\mathrm{b})$ intersects the curve $\mathrm{y}=\frac{\mathrm{c}^{2}}{\mathrm{x}}$ is $\left(\mathrm{x}_{1} \mathrm{y}_{\mathrm{i}}\right), \mathrm{i}=1,2,3,4 .$ Then
(a) A.M of all $x$ 's is zero
(b) A.M of all $\mathrm{y}_{i}$ 's is zero
(c) G.M of all $\mathrm{x}_{\mathrm{i}}$, is $|\mathrm{c}|$
(d) G.M of all $\mathrm{y}_{\mathrm{i}}^{\prime}$ is (c)

Goutam Chand
Goutam Chand
Numerade Educator
02:04

Problem 197

Equations of ellipse $E_{1}$ and $E_{2}$ are given by $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ and $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{a^{2}}=1(a>b)$. Then
(a) Director circles of $\mathrm{E}_{1}$ and $\mathrm{E}_{2}$ are the same
(b) Auxiliary circles of $E_{1}$ and $E_{2}$ are the same
(c) Circles, centre origin and passing through foci of $E_{1}$ and these of $E$, are the same
(d) Circle, centre origin passing through one of the points of intersection of $\mathrm{E}_{1}$ and $\mathrm{E}_{2}$ passes through all the points of intersection.

Harshita Goel
Harshita Goel
Numerade Educator
03:11

Problem 198

Match the elements of Column I to elements of Column II. There can be single or multiple matches.
The line $\mathrm{y}=\mathrm{mx}+\mathrm{c}$ is a tangent to
Column I
(a) the parabola $\mathrm{y}^{2}=4 \mathrm{x}$, if
(b) the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{4}=1$, if
(c) the hyperbola $\frac{x^{2}}{12}-\frac{y^{2}}{9}=1$, if
(d) the circle $x^{2}+y^{2}=3$, if
Column II
(p) $\mathrm{m}=2, \mathrm{c}=\pm \sqrt{15}$
(q) $\mathrm{m}=\frac{1}{\sqrt{5}}, \mathrm{c}=\pm 3$
(r) $\mathrm{m}=2, \mathrm{c}=\pm \sqrt{39}$
(s) $\quad \mathrm{m}=1, \mathrm{c}=1$

Charles Carter
Charles Carter
Numerade Educator
01:06

Problem 199

Match the elements of Column I to elements of Column II. There can be single or multiple matches.
The line $\mathrm{y}=\mathrm{mx}+\mathrm{c}$ is a tangent to
Column I
(a) Coordinates of the foci of the conic $25 x^{2}+16 y^{2}+50 x-192 y+201=0$
(b) Coordinates of the focus of the parabola $x^{2}-4 x-12 y-8=0$ are
(c) Coordinates of the centre of the director circle
of the circle $x^{2}+y^{2}-4 x-4 y-10=0$ are
(d) Coordinates of the focus of the parabola $\mathrm{y}^{2}+4 \mathrm{x}+12=0$ are
Column II
(p) $(2,2)$
(q) $(-1,3)$
(r) $(-4,0)$
(s) $(-1,9)$

AG
Ankit Gupta
Numerade Educator
09:45

Problem 200

Match the elements of Column I to elements of Column II. There can be single or multiple matches.
The line $\mathrm{y}=\mathrm{mx}+\mathrm{c}$ is a tangent to
Column I
(a) Radical axis of the circles $x^{2}+y^{2}-8 x-4 y-10=0$ and
$x^{2}+y^{2}-10 x-4 y-22=0$ is
(b) Equation of a common tangent to the circles $x^{2}+y^{2}+2 x-10 y+22=0$
$x^{2}+y^{2}-6 x-4 y+4=0$ is
(c) Equation of the directrix of the parabola $y^{2}-24 x=0$
(d) The parabola $x^{2}-4 y=0$ is touching the line
Column II
(p) $4 x-3 y+9=0$
(q) $2 x-y-4=0$
(r) $x+y+1=0$
(s) $x+6=0$

Harshita Goel
Harshita Goel
Numerade Educator