A Complete Resource Book in Mathematics for JEE Main

Dinesh Khattar

Chapter 18

Coordinates and Straight Lines - all with Video Answers

Educators

Chapter Questions

Problem 1

If one of the diagonals of a square is along the line $x=$ $y$ and one of its vertices is $(3,0)$, then its side through this vertex nearer to the origin is given by the equation.
(A) $y-3 x+9=0$
(B) $3 y+x-3=0$
(C) $x-3 y-3=0$
(D) $3 x+y-9=0$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 2

Through the point $P(\alpha, \beta)$, where $a \beta>0$ the straight line $\frac{x}{a}+\frac{y}{b}=1$ is drawn so as to form with coordinate axes a triangle of area $S$. If $a b>0$, then the least value of $S$ is
(A) $\alpha \beta$
(B) $2 \alpha \beta$
(C) $4 \alpha \beta$
(D) none of these

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 3

A line joining two points $A(2,0)$ and $B(3,1)$ is rotated about $A$ in anti-clockwise direction through an angle $15^{\circ} .$ If $B$ goes to $C$ in the new position, then the coordinates of $C$ are
(A) $\left(2, \sqrt{\frac{3}{2}}\right)$
(B) $\left(2,-\sqrt{\frac{3}{2}}\right)$
(C) $\left(2+\frac{1}{\sqrt{2}}, \sqrt{\frac{3}{2}}\right)$
(D) none of these

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 4

$P$ is a point on either of the two lines $y-\sqrt{3}|x|=2$ at a distance of 5 units from their point of intersection. The coordinates of the foot of the perpendicular from $P$ on the bisector of the angle between them are
(A) $\left[0, \frac{1}{2}(4+5 \sqrt{3})\right]$ or $\left[0, \frac{1}{2}(4-5 \sqrt{3})\right]$ depend-
ing on which line the point $P$ is taken
(B) $\left[0, \frac{1}{2}(4+5 \sqrt{3})\right]$
(C) $\left[0, \frac{1}{2}(4-5 \sqrt{3})\right]$
(D) $\left[\frac{5}{2}, \frac{5 \sqrt{3}}{2}\right]$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 5

A string of length 12 units is bent first into a square $P Q R S$ and then into a right-angled $\Delta P Q T$ by keeping the side $P Q$ of the square fixed and other is one more than its side. Then, the area of $P Q R S$ equals
(A) ar $(\Delta P Q T)$
(B) $\frac{3}{2} \cdot \operatorname{ar}(\Delta P Q T)$
(C) $2 \cdot \operatorname{ar}(\Delta P Q T)$
(D) none of these

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 6

The condition to be imposed on $\beta$ so that $(0, \beta)$ lies on or inside the triangle having sides $y+3 x+2=0$, $3 y-2 x-5=0$ and $4 y+x-14=0$ is
(A) $0<\beta<\frac{5}{3}$
(B) $0<\beta<\frac{7}{2}$
(C) $\frac{5}{3} \leq \beta \leq \frac{7}{2}$
(D) none of these

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 7

The point $(1, \beta)$ lies on or inside the triangle formed by the lines $y=x, x$-axis and $x+y=8$, if
(A) $0<\beta<1$
(B) $0 \leq \beta \leq 1$
(C) $0<\beta<8$
(D) none of these

Km Neeraj

Numerade Educator

Problem 8

A ray of light travelling along the line $x+\sqrt{3} y=5$ is incident on the $x$-axis and after refraction it enters the other side of the $x$-axis by turning $\frac{\pi}{6}$ away from the $x$-axis. The equation of the line along which the refracted ray travels is
(A) $x+\sqrt{3} y-5 \sqrt{3}=0$
(B) $x-\sqrt{3} y-5 \sqrt{3}=0$
(C) $\sqrt{3} x+y-5 \sqrt{3}=0$
(D) $\sqrt{3} x-y-5 \sqrt{3}=0$

Km Neeraj

Numerade Educator

Problem 9

A ray of light is sent along the line which passes through the point $(2,3)$. The ray is reflected from the point $P$ on $x$-axis. If the reflected ray passes through the point $(6,4)$, then the coordinates of $P$ are
(A) $\left(\frac{26}{7}, 0\right)$
(B) $\left(0, \frac{26}{7}\right)$
(C) $\left(-\frac{26}{7}, 0\right)$
(D) none of these

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 10

A line passing through the point $P(4,2)$, meets the $x$-axis and $y$-axis at $A$ and $B$, respectively. If $O$ is the origin, then locus of the centre of the circum circle of $\Delta O A B$ is
(A) $x^{-1}+y^{-1}=2$
(B) $2 x^{-1}+y^{-1}=1$
(C) $x^{-1}+2 y^{-1}=1$
(D) $2 x^{-1}+2 y^{-1}=1$

Km Neeraj

Numerade Educator

Problem 11

If the point $(2 \cos \theta, 2 \sin \theta)$ does not fall in that angle between the lines $y=|x-2|$ in which the origin lies then $\theta$ belongs to
(A) $\left(\frac{\pi}{2}, \frac{3 \pi}{2}\right)$
(B) $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$
(C) $(0, \pi)$
(D) none of these

Km Neeraj

Numerade Educator

Problem 12

If the equations of the sides of a triangle are $x+y=2$, $y=x$ and $\sqrt{3} y+x=0$, then which of the following is an exterior point of the triangle?
(A) orthocentre
(B) incentre
(C) centroid
(D) none of these

Km Neeraj

Numerade Educator

Problem 13

A line is drawn from the point $P(\alpha, \beta)$, making an angle $\theta$ with the positive direction of $x$-axis, to meet the line $a x+b y+c=0$ at $Q .$ The length of $P Q$ is
(A) $-\frac{a \alpha+b \beta+c}{a \cos \theta+b \sin \theta}$
(B) $\left|\frac{a \alpha+b \beta+c}{\sqrt{a^{2}+b^{2}}}\right|$
(C) $\frac{a \alpha+b \beta+c}{a \cos \theta+b \sin \theta}$
(D) none of these

Km Neeraj

Numerade Educator

Problem 14

If the equal sides $A B$ and $A C$ (each equal to $a$ ) of a right-angled isosceles triangle $A B C$ be produced to $P$ and $Q$ so that $B P \cdot C Q=A B^{2}$, then the line $P Q$ always passes through the fixed point
(A) $(a, 0)$
(B) $(0, a)$
(C) $(a, a)$
(D) none of these

Km Neeraj

Numerade Educator

Problem 15

If $x_{1}, x_{2}, x_{3}$ as well as $y_{1}, y_{2}, y_{3}$ are in G. P. with the same common ratio, then the points $\left(x_{1}, y_{1}\right),\left(x_{2}, y_{2}\right)$ and $\left(x_{3}, y_{3}\right)$
(A) lie on a straight line
(B) lie on an ellipse
(C) lie on a circle
(D) are vertices of a triangle

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 16

Number of equilateral triangles with $y=\sqrt{3}(x-1)+2$ and $y=-\sqrt{3} x$ as two of its sides, is
(A) 0
(B) 1
(C) 2
(D) none of these

Km Neeraj

Numerade Educator

Problem 17

If the distance of any point $P(x, y)$ from the origin is defined as $d(x, y)=\operatorname{Max} .\{|x|,|y|\}$ and $d(x, y)=k$ (nonzero constant), then the locus of the point $P$ is
(A) a straight line
(B) a circle
(C) a parabola
(D) none of these

Km Neeraj

Numerade Educator

Problem 18

If $a, b, c$ form an A. P. with common difference $d(\neq 0)$ and $x, y, z$ form a G. P. with common ratio $r(\neq 1)$, then the area of the triangle with vertices $(a, x),(b, y)$ and $(c, z)$ is independent of
(A) $b$
(B) $r$
(C) $d$
(D) $x$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 19

A line of fixed length 2 units moves so that its ends are on the positive $x$-axis and that part of the line $x+y=$ 0 which lies in the second quadrant. The locus of the mid-point of the line has the equation
(A) $(x+2 y)^{2}+y^{2}=1$
(B) $(x-2 y)^{2}+y^{2}=1$
(C) $(x+2 y)^{2}-y^{2}=1$
(D) none of these

Km Neeraj

Numerade Educator

Problem 20

A straight line through the origin $O$ meets the parallel lines $4 x+2 y=9$ and $2 x+y+6=0$ at points $P$ and $Q$, respectively. The point $O$ divides the segment $P Q$ in the ratio
(A) $1: 2$
(B) $3: 4$
(C) $2: 1$
(D) $4: 3$

P Krishnamurthy

P Krishnamurthy

Numerade Educator

Problem 21

Let $O$ be the origin and let $A(2,0), B(0,2)$ be two points. If $P(x, y)$ is a point such that $x y>0$ and $x+y<$ 2 , then
(A) $P$ lies either inside the triangle $O A B$ or in the third quadrant
(B) $P$ cannot be inside the triangle $O A B$
(C) $P$ lies inside the triangle $O A B$
(D) none of these

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 22

Consider the equation $y-y_{1}=m\left(x-x_{1}\right) .$ In this equation, if $m$ and $x_{1}$ are fixed and different lines are drawn for different values of $y^{1}$, then,
(A) the lines will pass through a single point
(B) there will be one possible line only
(C) there will be a set of parallel lines
(D) none of these

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 23

$D$ is a point on $A C$ of the triangle with vertices $A(2,$,
3), $B(1,-3), C(-4,-7)$ and $B D$ divides $A B C$ into two triangles of equal area. The equation of the line drawn through $B$ at right angles to $B D$ is
(A) $y-2 x+5=0$
(B) $2 y-x+5=0$
(C) $y+2 x-5=0$
(D) $2 y+x-5=0$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 24

If two points $A(a, 0)$ and $B(-a, 0)$ are stationary and if $\angle A-\angle B=\theta$ in $\triangle A B C$, the locus of $C$ is
(A) $x^{2}+y^{2}+2 x y \tan \theta=a^{2}$
(B) $x^{2}-y^{2}+2 x y \tan \theta=a^{2}$
(C) $x^{2}+y^{2}+2 x y \cot \theta=a^{2}$
(D) $x^{2}-y^{2}+2 x y \cot \theta=a^{2}$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 25

The straight line $y=x-2$ rotates about a point where it cuts the $x$-axis and becomes perpendicular to the straight line $a x+b y+c=0 .$ Then, its equation is
(A) $a x+b y+2 a=0$
(B) $a x-b y-2 a=0$
(C) $b y+a y-2 b=0$
(D) $a y-b x+2 b=0$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 26

If the point $P\left(a^{2}, a\right.$ ) lies in the region corresponding to the acute angle between the lines $2 y=x$ and $4 y=x$, then
(A) $a \in(2,6)$
(B) $a \in(4,6)$
(C) $a \in(2,4)$
(D) none of these

Km Neeraj

Numerade Educator

Problem 27

The point $(4,1)$ undergoes the following three successive transformations
(A) Reflection about the line $y=x-1$
(B) Translation through a distance 1 unit along the positive $x$-axis
(C) Rotation through an angle $\frac{\pi}{4}$ about the origin in the anti-clockwise direction. Then, the coordinates of the final point are
(A) $(4,3)$
(B) $\left(\frac{7}{2}, \frac{7}{2}\right)$
(C) $(0,3 \sqrt{2})$
(D) $(3,4)$

Km Neeraj

Numerade Educator

Problem 28

A light ray emerging from the point source placed at $P(2,3)$ is reflected at point ' $\theta$ ' on the $y$-axis and then passes through the point $R(5,10)$. Coordinates of ' $Q$ ' are
(A) $(0,3)$
(B) $(0,2)$
(C) $(0,5)$
(D) none of these

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 29

The distance between two parallel lines is unity. A point $P$ lies between the lines at a distance $a$ from one of them. The length of a side of an equilateral triangle $P Q R$, vertex $Q$ of which lies on one of the parallel lines and vertex $R$ lies on the other line, is
(A) $\frac{2}{\sqrt{3}} \cdot \sqrt{a^{2}+a+1}$
(B) $\frac{2}{\sqrt{3}} \sqrt{a^{2}-a+1}$
(C) $\frac{1}{\sqrt{3}} \sqrt{a^{2}+a+1}$
(D) $\frac{1}{\sqrt{3}} \sqrt{a^{2}-a+1}$

Km Neeraj

Numerade Educator

Problem 30

Two points $A$ and $B$ are given. $P$ is a moving point on one side of the line $A B$ such that $\angle P A B-\angle P B A$ is a positive constant $2 \theta$. The locus of the point $P$ is
(A) $x^{2}+y^{2}+2 x y \cot 2 \theta=a^{2}$
(B) $x^{2}+y^{2}-2 x y \cot 2 \theta=a^{2}$
(C) $x^{2}+y^{2}+2 x y \tan 2 \theta=a^{2}$
(D) $x^{2}-y^{2}+2 x y \cot 2 \theta=a^{2}$.

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 31

The four points $A(p, 0), B(q, 0), C(r, 0)$ and $D(s, 0)$ are such that $p, q$ are the roots of the equation $a x^{2}+2 h x+$ $b=0$ and $r, s$ are those of equation $a^{\prime} x^{2}+2 h^{\prime} x+b^{\prime}=0$. If the sum of the ratios in which $C$ and $D$ divide $A B$ is zero, then
(A) $a b^{\prime}+a^{\prime} b=2 h h^{\prime}$
(B) $a b^{\prime}+a^{\prime} b=h h^{\prime}$
(C) $a b^{\prime}-a^{\prime} b=2 h h^{\prime}$
(D) none of these

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 32

The coordinates of a point $P$ on the line $3 x+2 y+10$ $=0$ such that $|P A-P B|$ is maximum where $A$ is $(4,2)$ and $B$ is $(2,4)$, are
(A) $(22,28)$
(B) $(22,-28)$
(C) $(-22,28)$
(D) $(-22,-28)$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 33

A line through $A(-5,-4)$ meets the lines $x+3 y+2=0$, $2 x+y+4=0$ and $x-y-5=0$ at the point $B, C$ and $D$, respectively. If $\left(\frac{15}{A B}\right)^{2}+\left(\frac{10}{A C}\right)^{2}=\left(\frac{6}{A D}\right)^{2}$, the equa-
tion of the line is
(A) $2 x+3 y+22=0$
(B) $2 x-3 y+22=0$
(C) $3 x+2 y+22=0$
(D) $3 x-2 y+22=0$

Km Neeraj

Numerade Educator

Problem 34

$A(0,0), B(2,1)$ and $C(3,0)$ are the vertices of a $\triangle A B C$ and $B D$ is its altitude. If the line through $D$ parallel to the side $A B$ intersects the side $B C$ at a point $K$, then the product of the areas of the triangles $A B C$ and $B D K$ is
(A) 1
(B) $\frac{1}{2}$
(C) $\frac{1}{4}$
(D) none of these

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 35

A line cuts the $x$-axis at $A(7,0)$ and $y$-axis at $B(0,-5)$. A variable line $P Q$ is drawn $\perp$ to $A B$ cutting the $x$-axis in $P$ and the $y$-axis in $Q .$ If $A Q$ and $B P$ intersect at $R$, then the locus of $R$ is
(A) $x(x-7)+y(y+5)=0$
(B) $x(x-7)-y(y+5)=0$
(C) $x(x+7)+y(y-5)=0$
(D) none of these

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 36

The point $(2,3)$ undergoes the following three transformations successively
(i) reflection about the line $y=x$
(ii) translation through a distance 2 units along the positive direction of $y$-axis
(iii) rotation through an angle of $45^{\circ}$ about the origin in the anti-clockwise direction. The final coordinates of the point are
(A) $\left(\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)$
(B) $\left(-\frac{1}{\sqrt{2}}, \frac{7}{\sqrt{2}}\right)$
(C) $\left(\frac{1}{\sqrt{2}},-\frac{7}{\sqrt{2}}\right)$
(D) none of these

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 37

Lines $L_{1}=a x+b y+c=0$ and $L_{2}=l x+m y+n=0$
intersect at the point $P$ and make an angle $\theta$ with each other. The equation of line $L$ different from $L_{2}$ which passes through $\mathrm{P}$ and makes the same angle $\theta$ with $L_{1}$ is
(A) $2(a l+b m)(a x+b y+c)-\left(a^{2}+b^{2}\right)(l x+m y+n)$ $=0$
(B) $2(a l+b m)(a x+b y+c)+\left(a^{2}+b^{2}\right)(b x+m y+n)$ $=0$
(C) $2\left(a^{2}+b^{2}\right)(a x+b y+c)-(a l+b m)(l x+m y+n)$ $=0$
(D) none of these

Km Neeraj

Numerade Educator

Problem 38

The equations of the perpendicular bisector of the sides $A B$ and $A C$ of a $\Delta A B C$ are $x-y+5=0$ and $x+$ $2 y=0$, respectively. If the point $A$ is $(1,-2)$ then the equation of the line $B C$ is
(A) $14 x+23 y=40$
(B) $14 x-23 y=40$
(C) $23 x+14 y=40$
(D) $23 x-14 y=40$

Km Neeraj

Numerade Educator

Problem 39

The equation of a family of lines is given by $(2+3 t)$ $x+(1-2 t) y+4=0$, where $t$ is the parameter. The equation of a straight line, belonging to this family, at the maximum distance from the point $(2,3)$ is
(A) $21 x+14 y=0$
(B) $21 x-14 y=0$
(C) $14 x-21 y=0$
(D) none of these

Km Neeraj

Numerade Educator

Problem 40

$A B C D$ is a square whose vertices $A, B, C$ and $D$ are $(0,0),(2,0),(2,2)$ and $(0,2)$, respectively. This square is rotated in the $X-Y$ plane with an angle of $30^{\circ}$ in anti-clockwise direction about an axis passing through the vertex $A$. The equation of the diagonal $B D$ of this rotated square is
(A) $\sqrt{3} x+(1-\sqrt{3}) y=\sqrt{3}$
(B) $(1+\sqrt{3}) x-(1-\sqrt{2})=2$
(C) $(2-\sqrt{3}) x+y=2(\sqrt{3}-1)$
(D) none of these

Km Neeraj

Numerade Educator

Problem 41

The equations of the straight lines passing through $(-2,-7)$ and cutting an intercept of length three units between the straight lines $4 x+3 y=12$ and $4 x+3 y=$ 3 are
(A) $x+2=0, y+7=\frac{7}{24}(x+2)$
(B) $x-2=0, y+7=-\frac{7}{24}(x+2)$
(C) $x+2=0, y+7=-\frac{7}{24}(x+2)$
(D) $x+2=0, y+7=-\frac{7}{12}(x+2)$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 42

The coordinates of the point which is at unit distance from the lines $L_{1} \equiv 3 x-4 y+1=0$ and $L_{2} \equiv 8 x+6 y+$ $1=0$ and lies below $L_{1}$ and above $L_{2}$ are
(A) $\left(\frac{6}{5}, \frac{1}{10}\right)$
(B) $\left(\frac{6}{5},-\frac{1}{10}\right)$
(C) $\left(\frac{6}{5}, \frac{1}{5}\right)$
(D) $\left(\frac{6}{5},-\frac{1}{5}\right)$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 43

The vertices of a triangle are $A\left(x_{1}, x_{1} \tan \alpha\right), B\left(x_{2}, x_{2}\right.$ $\tan \beta$ ) and $C\left(x_{3}, x_{3} \tan \gamma\right)$. If the circumcentre of triangle $A B C$ coincides with the origin and $H(a, b)$ be its orthocentre then $\frac{a}{h}=$
(A) $\frac{\cos \alpha+\cos \beta+\cos \gamma}{\cos \alpha \cdot \cos \beta \cdot \cos \gamma}$
(B) $\frac{\sin \alpha+\sin \beta+\sin \gamma}{\sin \alpha \cdot \sin \beta \cdot \sin \gamma}$
(C) $\frac{\tan \alpha+\tan \beta+\tan \gamma}{\tan \alpha \cdot \tan \beta \cdot \tan \gamma}$
(D) $\frac{\cos \alpha+\cos \beta+\cos \gamma}{\sin \alpha+\sin \beta+\sin \gamma}$

Km Neeraj

Numerade Educator

Problem 44

$O X$ and $O Y$ are two coordinate axes. On $O Y$ is taken a fixed point $P$ and on $O X$ any point $Q .$ On $P Q$ an equilateral triangle is described, its vertex $R$ being on the side of $P Q$ away from $O$, then the locus of $R$ will be
(A) straight line
(B) circle
(C) ellipse
(D) parabola

Km Neeraj

Numerade Educator

Problem 45

If the vertices of a variable triangle are $(3,4),(5 \mathrm{cos}$ $\theta, 5 \sin \theta$ ) and $(5 \sin \theta,-5 \cos \theta)$, then the locus of its orthocentre is
(A) $(x+y-1)^{2}+(x-y-7)^{2}=100$
(B) $(x+y-7)^{2}+(x-y+1)^{2}=100$
(C) $(x+y-7)^{2}+(x-y-1)^{2}=100$
(D) $(x+y+7)^{2}+(x+y-1)^{2}=100$

Km Neeraj

Numerade Educator

Problem 46

If a right-angled isosceles triangle right-angled at origin has $3 x+4 y=6$ as its base, then the area of the triangle is
(A) 7
(B) $\frac{11}{25}$
(C) $\frac{36}{25}$
(D) $\frac{12}{25}$

Km Neeraj

Numerade Educator

Problem 47

The line $x+y=1$ meets $x$-axis at $A$ and $y$-axis at $B \cdot P$ is the mid-point of $A B \cdot P_{1}$ is the foot of the perpendicular from $P$ to $O A ; M_{1}$ is that from $P_{1}$ to $O P ; P_{2}$ is that from $M_{1}$ to $O A$ and so on. If $P_{n}$ denotes the foot of the $n$th perpendicular on $O A$ from $M_{n-1}$, then $O P_{n}$ is equal to
(A) $\frac{1}{2^{n}}$
(B) $\frac{1}{2^{n-1}}$
(C) $\frac{1}{2^{n-2}}$
(D) none of these

Km Neeraj

Numerade Educator

Problem 48

The line $x+y=a$ meets $x$-axis at $A$. A triangle $A M N$ is inscribed in the triangle $O A B, O$ being the origin with right angle at $N ; M$ and $N$ lie respectively on $O B$ and $A B$. If area of $\Delta A M N$ is $\frac{3}{8}$ of the area of triangle $O A B$, then $\frac{A N}{B N}$ is equal to
(A) 3
(B) $\frac{1}{3}$
(C) 2
(D) $\frac{2}{3}$

Km Neeraj

Numerade Educator

Problem 49

Let $S_{1}, S_{2}, \ldots$ be squares such that for each $n \geq 1$, the length of a side of $S_{n}$ equals the length of a diagonal of $S_{n+1} .$ If the length of a side of $S_{1}$ is $10 \mathrm{~cm}$, then for which of the following values of $n$ is the area of $S_{n}$ less than 1 square $\mathrm{cm} ?$
(A) 7
(B) 8
(C) 9
(D) 10

Km Neeraj

Numerade Educator

Problem 50

A line which makes an acute angle $\theta$ with the positive direction of $x$-axis is drawn through the point $P(3,4)$ to meet the line $x=6$ at $R$ and $y=8$ at $S$, then
(A) $P R=3 \sec \theta$
(B) $P S=4 \operatorname{cosec} \theta$
(C) $P R+P S=\frac{2(3 \sin \theta+4 \cos \theta)}{\sin 2 \theta}$
(D) $\frac{9}{(P R)^{2}}+\frac{16}{(P S)^{2}}=1$

Km Neeraj

Numerade Educator

Problem 51

Straight lines $3 x+4 y=5$ and $4 x-3 y=15$ intersect at A. Points $B$ and $C$ are choosen on these lines such that $A B=A C .$ The equation of the line $B C$ passing through the point $(1,2)$ is
(A) $x+7 y+13=0$
(B) $x-7 y+13=0$
(C) $7 x+y-9=0$
(D) none of these

Km Neeraj

Numerade Educator

Problem 52

The equation of the straight line passing through the point $(4,5)$ and making equal angles with the two straight lines given by the equations $3 x-4 y-7=0$ and $12 x-5 y+6=0$, is
(A) $9 x-7 y-1=0$
(B) $9 x+7 y-1=0$
(C) $7 x+9 y-73=0$
(D) $7 x+9 y+73=0$

Km Neeraj

Numerade Educator

Problem 53

Let the algebraic sum of the perpendicular distances from the points $A(2,0), B(0,2), C(1,1)$ to a variable line be zero. Then, all such lines
(A) are concurrent
(B) pass through the fixed point $(1,1)$
(C) touch some fixed circle
(D) pass through the centroid of $\triangle A B C$

Km Neeraj

Numerade Educator

Problem 54

The equation of the line passing through the point (2, 3) and making intercept of length 2 units between the lines $y+2 x=3$ and $y+2 x=5$, is
(A) $x=2$
(B) $3 x+4 y=18$
(C) $4 x+3 y=18$
(D) none of these

Km Neeraj

Numerade Educator

Problem 55

Two sides of a rhombus $A B C D$ are parallel to the lines $y=x+2$ and $y=7 x+3$. If the diagonals of the rhombus intersect at the point $(1,2)$ and the vertex $A$ is on the $y$-axis, then the possible coordinates of $A$ are
(A) $(0,0)$
(B) $\left(0, \frac{5}{2}\right)$
(C) $\left(0,-\frac{5}{2}\right)$
(D) none of these

Km Neeraj

Numerade Educator

Problem 56

The equations of two equal sides $A B$ and $A C$ of an isosceles triangle $A B C$ are $x+y=5$ and $7 x-y=3$, respectively. The equation of the side $B C$, if the area of $\triangle A B C$ is 5 units, is
(A) $3 x+y-2=0$
(B) $3 x+y-12=0$
(C) $x-3 y+1=0$
(D) $x-3 y+21=0$

Km Neeraj

Numerade Educator

Problem 57

If the equation of the mirror be $2 x+y-6=0$ and a ray passing through $(3,10)$ after being reflected by the mirror passes through $(7,2)$, then the equations of the incident ray and the reflected ray are
(A) $x+3 y-13=0$
(B) $3 x-y+1=0$
(C) $x-3 y+13=0$
(D) $3 x+y-1=0$

Km Neeraj

Numerade Educator

Problem 58

Line $x+2 y=4$ is translated by 3 units closer to the origin and then rotated by $30^{\circ}$ in the clockwise sence about the point where the shifted line cuts the $x$-axis. If the equation of the line in the new position is $y=m(x$ $+c$ ), then
(A) $m=\frac{2+\sqrt{3}}{2 \sqrt{3}-1}$
(B) $m=\frac{2+\sqrt{3}}{1-2 \sqrt{3}}$
(C) $c=3 \sqrt{5}-4$
(D) $c=4-3 \sqrt{5}$

Km Neeraj

Numerade Educator

Problem 59

In oblique coordinates, the equation $y=m x+c$ represents a straight line which is inclined at an angle $$ \tan ^{-1}\left(\frac{m \sin w}{1+m \cos w}\right) $$ to the $x$-axis, where $w$ is the angle between the axes. If $\theta$ be the angle between two lines $y=m_{1} x+c_{1}$ and $y=m_{2} x$ $+c_{2}, w$ be the angle between the axes, then $$ \tan \theta=\frac{\left(m_{1}-m_{2}\right) \sin w}{1+\left(m_{1}+m_{2}\right) \cos w+m_{1} m_{2}} $$ The two given lines are parallel if $m_{1}=m_{2}$. The two lines are perpendicular if $1+\left(m_{1}+m_{2}\right) \cos w+$ $m_{1} m_{2}=0$
If the straight lines $y=m_{1} x+c_{1}$ and $y=m_{2} x+c_{2}$ make equal angles with the axis of $x$ and be not parallel to one another, then $m_{1}+m_{2}+k m_{1} m_{2} \cos w=0$ where $k=$
(A) 1
(B) 2
(C) $-1$
(D) $-2$

Km Neeraj

Numerade Educator

Problem 60

In oblique coordinates, the equation $y=m x+c$ represents a straight line which is inclined at an angle $$ \tan ^{-1}\left(\frac{m \sin w}{1+m \cos w}\right) $$ to the $x$-axis, where $w$ is the angle between the axes. If $\theta$ be the angle between two lines $y=m_{1} x+c_{1}$ and $y=m_{2} x$ $+c_{2}, w$ be the angle between the axes, then $$ \tan \theta=\frac{\left(m_{1}-m_{2}\right) \sin w}{1+\left(m_{1}+m_{2}\right) \cos w+m_{1} m_{2}} $$ The two given lines are parallel if $m_{1}=m_{2}$. The two lines are perpendicular if $1+\left(m_{1}+m_{2}\right) \cos w+$ $m_{1} m_{2}=0$
The axes being inclined at an angle of $30^{\circ}$, the slope of the line which passes through the point $(-2,3)$ and is perpendicular to the straight line $y+3 x=6$ is
(A) $\frac{3 \sqrt{3}-2}{\sqrt{3}-6}$
(B) $\frac{3 \sqrt{3}+2}{\sqrt{3}-6}$
(C) $\frac{3 \sqrt{3}-2}{\sqrt{3}+6}$
(D) none of these

Km Neeraj

Numerade Educator

Problem 61

In oblique coordinates, the equation $y=m x+c$ represents a straight line which is inclined at an angle $$ \tan ^{-1}\left(\frac{m \sin w}{1+m \cos w}\right) $$ to the $x$-axis, where $w$ is the angle between the axes. If $\theta$ be the angle between two lines $y=m_{1} x+c_{1}$ and $y=m_{2} x$ $+c_{2}, w$ be the angle between the axes, then $$ \tan \theta=\frac{\left(m_{1}-m_{2}\right) \sin w}{1+\left(m_{1}+m_{2}\right) \cos w+m_{1} m_{2}} $$ The two given lines are parallel if $m_{1}=m_{2}$. The two lines are perpendicular if $1+\left(m_{1}+m_{2}\right) \cos w+$ $m_{1} m_{2}=0$
If $y=x \tan \frac{11 \pi}{24}$ and $y=x \tan \frac{19 \pi}{24}$ represent two
straight lines at right angles, then the angle between the axes is
(A) $\frac{\pi}{6}$
(B) $\frac{\pi}{4}$
(C) $\frac{\pi}{3}$
(D) $\frac{\pi}{2}$

Km Neeraj

Numerade Educator

Problem 62

In oblique coordinates, the equation $y=m x+c$ represents a straight line which is inclined at an angle $$ \tan ^{-1}\left(\frac{m \sin w}{1+m \cos w}\right) $$ to the $x$-axis, where $w$ is the angle between the axes. If $\theta$ be the angle between two lines $y=m_{1} x+c_{1}$ and $y=m_{2} x$ $+c_{2}, w$ be the angle between the axes, then $$ \tan \theta=\frac{\left(m_{1}-m_{2}\right) \sin w}{1+\left(m_{1}+m_{2}\right) \cos w+m_{1} m_{2}} $$ The two given lines are parallel if $m_{1}=m_{2}$. The two lines are perpendicular if $1+\left(m_{1}+m_{2}\right) \cos w+$ $m_{1} m_{2}=0$
The axes being inclined at an angle of $120^{\circ}$, the tangent of the angle between the two straight lines $8 x+7 y$ $=1$ and $28 x-73 y=101$ is $\tan ^{-1} \theta$, where $\theta=$
(A) $\frac{30 \sqrt{3}}{37}$
(B) $\frac{15 \sqrt{3}}{37}$
(C) $\frac{7 \sqrt{3}}{37}$
(D) none of these

Km Neeraj

Numerade Educator

Problem 63

Column-I Column-II
I. The diagonals of the parallelogram (A) $\frac{5 \pi}{12}$
whose sides are $l x+m y+n=0, l x+$ $m y+n=0, m x+l y+n=0, m x+l y+$
$n^{\prime}=0$ include an angle
II. The line $x+y=2$ turns about the point
(B) $\frac{\pi}{12}$
on it, whose ordinate is equal to abscissa, through an angle $\theta$ in the clockwise direction so that its equation becomes $y$ $=2 x-1$. Then, the value of the angle $\theta$ is
III. The larger of the two angles made with the $x$-axis of a straight line drawn (C) $\frac{\pi}{2}$ through $(1,2)$ so that it intersects $x+y$ $=4$ at a point distant $\frac{\sqrt{6}}{3}$ from $(1,2)$ is (D) $\tan ^{-1} 3$

Km Neeraj

Numerade Educator

Problem 64

Column-I Column-II I
I. If the points $A(x, y, z), B(y, z+x)$ and (A) $C(z, x+y)$ are such that $A B=B C$, then $x, y, z$ are in
II. If a line through the variable point (B) G.P. $A(k+1,2 k)$ meets the lines $7 x+y-$ $16=0,5 x-y-8=0, x-5 y+8=0$
at $B, C$ and $D$, respectively, then $A C$, $A B$ and $A D$ are in
III. The length of the perpendiculars
(C) A.G.P. from the points $\left(m^{2}, 2 m\right),(m n, m+n)$ and $\left(n^{2}, 2 n\right)$ to the line $x \cos \theta+y \sin \theta$
$=p$, where $p=-\frac{\sin ^{2} \theta}{\cos \theta}$, form a
IV. If the lines $a x+12 y+1=0, b x+$ (D) A.P. $13 y+1=0$ and $c x+14 y+1=0$ are
concurrent, then $a, b, c$ are in

Km Neeraj

Numerade Educator

Problem 65

A triangle with vertices $(4,0),(-1,-1),(3,5)$ is:
(A) isosceles and right angled
(B) isosceles but not right angled
(C) right angled but not isosceles
(D) neither right angled nor isosceles

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 66

The equation of the directrix of the parabola $y^{2}+4 y+$ $4 x+2=0$ is:
(A) $x=-1$
(B) $x=1$
(C) $x=-\frac{3}{2}$
(D) $x=\frac{3}{2}$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 67

The incentre of the triangle with vertices $(1, \sqrt{3}),(0,$, 0) and $(2,0)$ is:
(A) $\left(1, \frac{\sqrt{3}}{2}\right)$
(B) $\left(\frac{2}{3}, \frac{1}{\sqrt{3}}\right)$
(C) $\left(\frac{2}{3}, \frac{\sqrt{3}}{2}\right)$
(D) $\left(1, \frac{1}{\sqrt{3}}\right)$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 68

Three straight lines $2 x+11 y-5=0,24 x+7 y-20=$ 0 and $4 x-3 y-2=0$ :
(A) form a triangle
(B) are only concurrent
(C) are concurrent with one line bisecting the angle between the other two
(D) none of the above

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 69

A straight line through the point $(2,2)$ intersects the lines $\sqrt{3} x+y=0$ and $\sqrt{3} x-y=0$ at the points $A$ and $B$. The equation to the line $A B$ so that the triangle $O A B$ is equilateral, is:
(A) $x-2=0$
(B) $y-2=0$
(C) $x+y-4=0$
(D) none of these

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 70

If the equation of the locus of a point equidistant from the points $\left(a_{1}, b_{1}\right)$ and $\left(a_{2}, b_{2}\right)$ is $\left(a_{1}-a_{2}\right) x+\left(b_{1}-b_{2}\right) y$ $+c=0$, then the value of $^{\circ} c^{\prime}$ is $\quad$
(A) $\frac{1}{2}\left(a_{2}^{2}+b_{2}^{2}-a_{1}^{2}-b_{1}^{2}\right)$
(B) $a_{1}^{2}+a_{2}^{2}-b_{1}^{2}-b_{2}^{2}$
(C) $\frac{1}{2}\left(a_{1}^{2}+a_{2}^{2}-b_{1}^{2}-b_{2}^{2}\right)$
(D) $\sqrt{a_{1}^{2}+b_{1}^{2}-a_{2}^{2}-b_{2}^{2}}$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 71

Locus of centroid of the triangle whose vertices are (a $\cos t, a \sin t),(b \sin t,-b \cos t)$ and $(1,0)$, where $t$ is a
parameter, is
(A) $(3 x-1)^{2}+(3 y)^{2}=a^{2}-b^{2}$
(B) $(3 x-1)^{2}+(3 y)^{2}=a^{2}+b^{2}$
(C) $(3 x+1)^{2}+(3 y)^{2}=a^{2}+b^{2}$
(D) $(3 x+1)^{2}+(3 y)^{2}=a^{2}-b^{2}$

Km Neeraj

Numerade Educator

Problem 72

Let $A(2,-3)$ and $B(-2,1)$ be vertices of a triangle $A B C$. If the centroid of this triangle moves on the line $2 x+$ $3 y=1$, then the locus of the vertex $C$ is the line
(A) $2 x+3 y=9$
(B) $2 x-3 y=7$
(C) $3 x+2 y=5$
(D) $3 x-2 y=3$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 73

The equation of the straight line passing through the point $(4,3)$ and making intercepts on the co-ordinate axes whose sum is $-1$ is
(A) $\frac{x}{2}+\frac{y}{3}=-1$ and $\frac{x}{-2}+\frac{y}{1}=-1$
(B) $\frac{x}{2}-\frac{y}{3}=-1$ and $\frac{x}{-2}+\frac{y}{1}=-1$
(C) $\frac{x}{2}+\frac{y}{3}=1$ and $\frac{x}{2}+\frac{y}{1}=1$
(D) $\frac{x}{2}-\frac{y}{3}=1$ and $\frac{x}{-2}+\frac{y}{1}=1$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 74

If the sum of the slopes of the lines given by $x^{2}-$ $2 c x y-7 y^{2}=0$ is four times their product, then $c$ has the value
(A) 1
(B) $-1$
(C) 2
(D) $-2$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 75

If one of the lines given by $6 x^{2}-x y+4 c y^{2}=0$ is $3 x+$ $4 y=0$, then $c$ equals
(A) 1
(B) $-1$
(C) 3
(D) $-3$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 76

Let $P$ be the point $(1,0)$ and $Q$ a point on the locus $y^{2}$ $=8 x$. The locus of mid-point of $P Q$ is
(A) $y^{2}-4 x+2=0$
(B) $y^{2}+4 x+2=0$
(C) $x^{2}+4 y+2=0$
(D) $x^{2}-4 y+2=0$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 77

The line parallel to the $x$-axis and passing through the intersection of the lines $a x+2 b y+3 b=0$ and $b x-2 a y$ $-3 a=0$, where $(a, b) \neq(0,0)$ is
(A) below the $x$-axis at a distance of $\frac{3}{2}$ from it
(B) below the $x$-axis at a distance of $\frac{2}{3}$ from it
(C) above the $x$-axis at a distance of $\frac{3}{2}$ from it
(D) above the $x$-axis at a distance of $\frac{2}{3}$ from it

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 78

If a vertex of a triangle is $(1,1)$ and the mid-points of two sides through this vertex are $(-1,2)$ and $(3,2)$ then the centroid of the triangle is
(A) $\left(-1, \frac{7}{3}\right)$
(B) $\left(\frac{-1}{3}, \frac{7}{3}\right)$
(C) $\left(1, \frac{7}{3}\right)$
(D) $\left(\frac{1}{3}, \frac{7}{3}\right)$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 79

A straight line through the point $A(3,4)$ is such that its intercept between the axes is bisected at $A$. Its equation is
(A) $x+y=7$
(B) $3 x-4 y+7=0$
(C) $4 x+3 y=24$
(D) $3 x+4 y=25$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 80

The locus of the vertices of the family of parabolas $y=\frac{a^{3} x^{2}}{3}+\frac{a^{2} x}{2}-2 a$ is
(A) $x y=\frac{105}{64}$
(B) $x y=\frac{3}{4}$
(C) $x y=\frac{35}{16}$
(D) $x y=\frac{64}{105}$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 81

If $\left(a, a^{2}\right)$ falls inside the angle made by the lines $y=\frac{x}{2}$, $x>0$ and $y=3 x, x>0$, then $a$ belongs to
(A) $\left(0, \frac{1}{2}\right)$
(B) $(3, \infty)$
(C) $\left(\frac{1}{2}, 3\right)$
(D) $\left(-3,-\frac{1}{2}\right)$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 82

Let $A(h, k), B(1,1)$ and $C(2,1)$ be the vertices of a right angled triangle with $A C$ as its hypotenuse. If the area of the triangle is 1 , then the set of values which ' $k$ ' can take is given by
(A) $\{1,3\}$
(B) $\{0,2\}$
(C) $\{-1,3\}$
(D) $\{-3,-2\}$

Km Neeraj

Numerade Educator

Problem 83

Let $P=(-1,0), Q=(0,0)$ and $R=(3,3 \sqrt{3})$ be three points. The equation of the bisector of the angle $P Q R$
(A) $\sqrt{3} x+y=0$
(B) $x+\frac{\sqrt{3}}{2} y=0$
(C) $\frac{\sqrt{3}}{2} x+y=0$
(D) $x+\sqrt{3} y=0$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 84

If one of the lines of $m y^{2}+\left(1-m^{2}\right) x y-m x^{2}=0$ is a bisector of the angle between the lines $x=0$ and $y=0$, then $m$ is
(A) $-\frac{1}{2}$
(B) $-2$
(C) 1
(D) 2

P Krishnamurthy

P Krishnamurthy

Numerade Educator

Problem 85

The perpendicular bisector of the line segment joining $P(1,4)$ and $Q(k, 3)$ has $y$-intercept $-4$. Then a possible value of $k$ is
(A) 1
(B) 2
(C) $-2$
(D) $-4$

Km Neeraj

Numerade Educator

Problem 86

The line $L$ given by $\frac{x}{5}+\frac{y}{b}=1$ passes through the point (13, 32) and the line $K$ which is parallel to $L$ has the equation $\frac{x}{c}+\frac{y}{3}=1 .$ Then, the distance between $L$ and $K$ is $c^{3}$
(A) $\sqrt{17}$
(B) $\frac{17}{\sqrt{15}}$
(C) $\frac{23}{\sqrt{17}}$
(D) $\frac{23}{\sqrt{15}}$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 87

The lines $L_{1}: y-x=0$ and $L_{2}: 2 x+y=0$ intersect the line $L_{3}: y+2=0$ at two respective points $P$ and $Q$. The bisector of the acute angle between $L_{1}$ and $L_{2}$ intersect $L_{3}$ at $R$.
Statement - $1:$ The ratio $P R: R Q$ equals $2 \sqrt{2}: \sqrt{5}$. Statement - $2:$ In any triangle, bisector of an angle divides the triangle into two similar triangles.
(A) Statement - 1 is true, Statement- 2 is true; Statement $-2$ is not a correct explanation for Statement $-1$
(B) Statement - 1 is true, Statement- 2 is false.
(C) Statement - 1 is false, Statement- 2 is true.
(D) Statement $-1$ is true, Statement $-2$ is true; Statement $-2$ is a correct explanation for Statement $-1$

Km Neeraj

Numerade Educator

Problem 88

Equation of the ellipse which passes through the point $(-3,1)$, whose axes are the coordinate axes and has eccentricity $\sqrt{\frac{2}{5}}$ is
(A) $5 x^{2}+3 y^{2}-48=0$
(B) $3 x^{2}+5 y^{2}-15=0$
(C) $5 x^{2}+3 y^{2}-32=0$
(D) $3 x^{2}+5 y^{2}-32=0$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 89

If the line $2 x+y=k$ passes through the point which divides the line segment joining the points $(1,1)$ and $(2,4)$ in the ratio $3: 2$, then $k$ equals
(A) $\frac{29}{5}$
(B) 5
(C) 6
(D) $\frac{11}{5}$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 90

A line is drawn through the point $(1,2)$ to meet the coordinate axes at points $P$ and $Q$ respectively such that it forms a triangle $O P Q$, where $O$ is the origin. If the area of the triangle $O P Q$ is least, then the slope of the line $P Q$ is
(A) $-\frac{1}{4}$
(B) $-4$
(C) $-2$
(D) $-\frac{1}{2}$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 91

A ray of light along $x+\sqrt{3 y}=\sqrt{3}$ gets reflected upon reaching $x$-axis, the equation of the reflected ray is
(A) $\sqrt{3 y}=x-\sqrt{3}$
(B) $y=\sqrt{3 x}-\sqrt{3}$
(C) $\sqrt{3 y}=x-1$
(D) $y=x+\sqrt{3}$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 92

The abscissa of the incentre of the triangle that has the coordinates of mid points of its sides as $(0,1)(1,1)$ and $(1,0)$ is
(A) $2-\sqrt{2}$
(B) $1+\sqrt{2}$
(C) $1-\sqrt{2}$
(D) $2+\sqrt{2}$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 93

Let $a, b, c$ and $d$ be non-zero numbers. If the point of intersection of the line $4 a x+2 a y+c=0$ with the line $5 b x+2 b y+d=0$ lies in the fourth quadrant and is equidistant from the two axes then
(A) $2 b c-3 a d=0$
(B) $2 b c+3 a d=0$
(C) $3 b c-2 a d=0$
(D) $3 b c+2 a d=0$

Km Neeraj

Numerade Educator

Problem 94

Let $P S$ be the median of the triangle with vertices $P$ $(2,2), Q(6,-1)$ and $R(7,3)$. The equation of the line passing through $(1,-1)$ and parallel to $P S$ is
(A) $4 x-7 y-11=0$
(B) $2 x+9 y+7=0$
(C) $4 x+7 y+3=0$
(D) $2 x-9 y-11=0$

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 95

The number of points, having both co-ordinates as integers, which lie in the interior of the triangle with vertices $(0,0),(0,41)$ and $(41,0)$, is:
(A) 861
(B) 820
(C) 780
(D) 901

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 96

Locus of the image of the point $(2,3)$ in the line $(2 x-$ $3 y+4)+k(x-2 y+3)=0, k \in R$, is a:
(A) straight line parallel to $y$-axis.
(B) circle of radius 2 .
(C) circle of radius 3 .
(D) straight line parallel to $x$-axis.

Mahipal Kumawat

Mahipal Kumawat

Numerade Educator

Problem 97

Two sides of a rhombus are along the lines, $x-y+1$ $=0$ and $7 x-y-5=0 .$ If its diagonals intersect at $(-1,$, $-2$ ), then which one of the following is a vertex of this rhombus?
(A) $\left(-\frac{10}{3},-\frac{7}{3}\right)$
(B) $(-3,-9)$
(C) $(-3,-8)$
(D) $\left(\frac{1}{3},-\frac{8}{3}\right)$

Km Neeraj

Numerade Educator