Differential Calculus Booster with Problems and Solutions for JEE Main and Advanced

Rejaul Makshud

Chapter 8 The Tangent and Normal - all with Video Answers

Educators

Section 1

Level I

01:05

Problem 1

Find the slope of the tangent to the curve $y=x^{3}+3 x^{2}+3 x-10$ at $x=2$.

Varsha Aggarwal

Numerade Educator

01:17

Problem 2

Find the slope of the normal to the curve $y=x^{x}+1$ at $x=2$.

Varsha Aggarwal

Numerade Educator

01:26

Problem 3

If the slope of the curve $y=\frac{a x}{b-x}$ at the point $(1,1)$ be 2, find the value of $a+b+10$.

Varsha Aggarwal

Numerade Educator

01:03

Problem 4

Find the equation of the tangent to the curve $y=e^{2 x}$ at $x=0$.

Varsha Aggarwal

Numerade Educator

01:00

Problem 5

Find the equation of the tangent to the curve $y=\sqrt[3]{x-1}$ at $x=1$.

Varsha Aggarwal

Numerade Educator

01:15

Problem 6

Find the equation of the tangent to the curve $y=b e^{-x / a}$ at the point where it crosses the $y$-axis.

Varsha Aggarwal

Numerade Educator

01:14

Problem 7

Find the equation of the tangent to the curve
$y=\frac{4}{x^{2}+2}$ at $x=0 .$.

Varsha Aggarwal

Numerade Educator

01:17

Problem 8

Find the equation of the normal to the curve $x^{3}+y^{3}=6 x y$ at $(3,3)$.

Varsha Aggarwal

Numerade Educator

02:41

Problem 9

Find the equation of the normal to the curve $y=3 x^{2}+2 \sin x+4 \cos x+10$ at $x=0$.

Charles Machakwa

Numerade Educator

01:18

Problem 10

Find the equation of the normal to the curve $x+y=x^{y}$, where it cuts the $x$-axis.

Varsha Aggarwal

Numerade Educator

01:44

Problem 11

Find the equation of the normal to the curve
$y=\mid x^{2}-\lfloor x \|$ at $x=-2$.

Varsha Aggarwal

Numerade Educator

02:39

Problem 12

Find the equations of the tangents drawn to the curve
$y^{2}-2 x^{3}-4 y+8=0$ from point $(1,2)$.

Varsha Aggarwal

Numerade Educator

01:38

Problem 13

Find the equation of the normal to the curve $x^{2}=4 y$, which passes through the point $(1,2)$

Dharmendra Jain

Numerade Educator

01:15

Problem 14

Show that the curve
$y=x-e^{x y}$ has a vertical tangent at $(1,0)$.

Varsha Aggarwal

Numerade Educator

01:25

Problem 15

The curve $x+y-\log (x+y)=2 x+5$ has a vertical tangent at the point $(p, q)$, then find the value of $p+q+10 .$

Varsha Aggarwal

Numerade Educator

06:56

Problem 16

Find the points on the curve $y=2 x^{3}+3 x^{2}-12 x+1$, where the tangent is horizontal.

Kira Schwander

Numerade Educator

01:25

Problem 17

Find the points on the curve $y=x^{3}-x^{2}-x+3$, where the tangents are parallel to $x$-axis.

Varsha Aggarwal

Numerade Educator

01:42

Problem 18

At what points on the curve $y=\frac{x^{3}}{3}+\frac{x^{2}}{2}$, the tangents make equal angles with the co-ordinate axes?

Varsha Aggarwal

Numerade Educator

01:07

Problem 19

Find the equation of the tangent to the curve $y=4 a x$ at ' $t^{\prime}$.

Varsha Aggarwal

Numerade Educator

01:13

Problem 20

Find the equation of the tangent to the curve $x^{2}+y^{2}+x+y=0$ at $(1,-1)$

Varsha Aggarwal

Numerade Educator

00:54

Problem 21

Find the equation of the normal to the curve $x^{2}+y^{2}=10$ at $(3,1)$.

Varsha Aggarwal

Numerade Educator

01:21

Problem 22

Find the equation of the normal to the curve
$x^{2}+y^{2}+4 x+6 y+9=0$ at $(-4,-3)$..

Varsha Aggarwal

Numerade Educator

01:38

Problem 23

Find the equation of the normal to the curve $\frac{x^{2}}{9}+\frac{y^{2}}{4}=2$ at $(3,2)$.

Varsha Aggarwal

Numerade Educator

01:21

Problem 24

Find the number of tangents to the curve $y^{2}-2 x^{2}-4 y+8=0$, which pass through the point $(1,2)$.

Varsha Aggarwal

Numerade Educator

01:53

Any tangent at a point $P(x, y)$ to the ellipse $\frac{x^{2}}{8}+\frac{y^{2}}{18}=1$ meets the co-ordinate axes in the points $A$ and $B$ such that the area of the triangle $\mathrm{OAB}$ is least, then find the point $P$.

Shima Shaw

Numerade Educator

01:59

Problem 26

If the tangent to the curve $\sqrt{x}+\sqrt{y}=\sqrt{a}$ at any point on it cuts the axes OX, OY respectively, then prove that $\mathrm{OP}+\mathrm{OQ}=a$.

Shima Shaw

Numerade Educator

02:11

Problem 27

Show that the $x$-intercept of the tangent at an arbitrary point of the curve $\frac{a}{x^{2}}+\frac{b}{y^{2}}=1$ is proportional to the cube of the abscissa of the point of tangency.

Varsha Aggarwal

Numerade Educator

03:21

Problem 28

Find the tangent to the curve $x^{3}+3 x y+y^{3}+x^{2} y=0$ at the origin.

Charles Machakwa

Numerade Educator

01:51

Problem 29

Find the tangent to the curve $a x^{2}+2 h x y+b y^{2}+a x$ $+b y=0$ at the origin.

Charles Machakwa

Numerade Educator

02:24

Problem 30

Find the tangent to the curve $\left(x^{4}+y^{4}\right)^{2}=2013\left(x^{2}-y^{2}\right)$ at the origin.

Charles Machakwa

Numerade Educator

03:17

Problem 31

Find the tangent to the curve $x^{5}+y^{5}+2010 x^{2}$ $-2011 y^{2}+2012 x-2013 y=0$ at the origin.

Charles Machakwa

Numerade Educator

03:15

Problem 32

Find the angle of intersection of the curves $x^{2}=y$ and $y^{2}=x$.

Varsha Aggarwal

Numerade Educator

01:58

Problem 33

Find the angle of intersection of the curves $y=4-x^{2}$ and $y=x^{2}$.

Varsha Aggarwal

Numerade Educator

02:01

Problem 34

Find the angle between the curves $y^{2}=4 x$ and $y=e^{-x / 2}$

Varsha Aggarwal

Numerade Educator

01:52

Problem 35

Find the acute angle between the curves
$y=\sin x$ and $y=\cos x$

Varsha Aggarwal

Numerade Educator

02:05

Problem 36

Find the angle between the curves
$2 y^{2}=x^{3}$ and $y^{2}=32 x$.

Shima Shaw

Numerade Educator

01:51

Problem 37

Prove that the curves $y^{2}=4 x$ and $x^{2}+y^{2}-6 y+1=0$ touch each other at the point $(1,2)$.

Varsha Aggarwal

Numerade Educator

01:53

Problem 38

Prove that the curves $y=6-x+x^{2}$ and $y=(x-1)(x+2)$ touch each other at $(2,4)$.

Varsha Aggarwal

Numerade Educator

02:15

Problem 39

Prove that the curves $x=y$ and $x y=k$ cut at right angles if $8 k^{2}=1$.

Varsha Aggarwal

Numerade Educator

02:13

Problem 40

Find the values of $a$ if the curves $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{4}=1$ and $y^{3}=16 x$ cut each other orthogonally.

Varsha Aggarwal

Numerade Educator

02:36

Problem 41

Find the acute angle between the curves $y=\left|x^{2}-1\right|$ and $y=\left\lfloor x^{2}-3 \mid\right.$ at their points of intersection.

Varsha Aggarwal

Numerade Educator

01:51

Problem 42

Find the angle of intersection of the curves
$y=[\operatorname{lsin} x|+| \cos x \mid]$, where $[,]=$, G.I.F and $x^{2}+y^{2}=5$.

Varsha Aggarwal

Numerade Educator

01:21

Problem 43

Find the shortest distance between the line $y=x-2$ and the curve $y=x^{2}+3 x+2$

Varsha Aggarwal

Numerade Educator

01:33

Problem 44

Find the shortest distance between the curves
$y^{2}=4 x$ and $x^{2}+y^{2}-12 x+31=0 .$

Shima Shaw

Numerade Educator

03:22

Problem 45

Find the shortest distance between the curves
$y^{2}=x^{3}$ and $9 x^{2}+9 y^{2}-30 y+16=0 .$

Varsha Aggarwal

Numerade Educator

01:31

Problem 46

Find a point on the curve $x^{2}+2 y^{2}=6$ whose distance from the line $x+y=7$ is minimum.

Shima Shaw

Numerade Educator

02:02

Problem 47

Find the least distance between any two points of the curves $y=3^{x}$ and $\log _{3} x$.

Shima Shaw

Numerade Educator

02:30

Problem 48

Find the common tangent to the curves $y=x^{2}+x+1$ and $y=x^{2}-5 x+6 .$

Varsha Aggarwal

Numerade Educator

01:35

Problem 49

Find the equation of the common tangent to the curves $y=3 x^{2}$ and $y=2 x^{3}+1$.

Varsha Aggarwal

Numerade Educator

01:34

Problem 50

Find the equation of the common tangent to the curves $y=6-x-x^{2}$ and $x y=x+3$

Varsha Aggarwal

Numerade Educator

02:27

Problem 51

If the tangent at $P(1,1)$ to the curve $y^{2}=x(2-x)^{2}$ meets the curve again at $Q$, then find the co-ordinates of $Q$.

Varsha Aggarwal

Numerade Educator

03:46

Problem 52

If the tangent at $P$ to the curve $y^{2}=x^{3}$ intersects the curve again at $Q$ and the straight line $O P, O Q$ makes angles $\alpha, \beta$ with the $x$-axis, where $O$ is origin, then find the value of $\left(\frac{\tan \alpha}{\tan \beta}+2013\right)$.

Gregory Higby

Numerade Educator

01:46

Problem 53

If the tangent at a variable point $P$ on the curve $y=x^{2}-x^{3}$ meets it again at $Q$. Show that the locus of the middle point of $P Q$ is $y=1-9 x+28 x^{2}-28 x^{3}$.

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Numerade Educator

03:51

Problem 54

A curve is given by the equations $x=\sec ^{2} \theta$ and $y=\cot \theta .$ If the tangent at $\mathrm{P}$, where $\theta=\frac{\pi}{4}$ meets the curve again at $\mathrm{Q}$. Find the length of $P Q$.

Shima Shaw

Numerade Educator

01:52

Problem 55

Find the lengths of the tangent, sub-tangent, normal and sub-normal to the curve $y^{2}=4 a x$ at the point $P\left(\mathrm{at}^{2}, 2 \mathrm{at}\right)$.

Varsha Aggarwal

Numerade Educator

01:02

Problem 56

Prove that the length of the sub-tangent at any point to the curve $y=b e^{x / a}$ is always constant.

Varsha Aggarwal

Numerade Educator

02:13

Problem 57

Find the length of the tangent and normal to the curves $x=a(\theta-\sin \theta), y=a(1-\cos \theta)$ at the point,
where $\theta=\frac{\pi}{2}$.

Varsha Aggarwal

Numerade Educator

00:46

Problem 58

Find the length of the sub-normal to the curve $y^{2}=x^{3}$ at $(4,8)$.

Varsha Aggarwal

Numerade Educator

01:19

Problem 59

Find the length of the sub-tangent to the curve
$y=\frac{c}{2}\left(e^{\frac{x}{c}}+e^{-\frac{x}{c}}\right)$ at any point $(x, y) .$

Shima Shaw

Numerade Educator

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