Tangents, Normals and Rate Measure

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JEE Main 2016

LEVELJEE Main

Consider: $f (x) = tan^{- 1} (\frac{1 + s i n x}{1 - s i n x}), x \in (0, \frac{π}{2})$ . A normal to $y = f (x)$ at $x = \frac{π}{6}$ also passes through the point

The normal to the curve, $x^{2} + 2 x y - 3 y^{2} = 0$ , at $(1, 1)$

(A)

meets the curve again in the third quadrant.

(B)

meets the curve again in the fourth quadrant.

(C)

does not meet the curve again.

(D)

meets the curve again in the second quadrant.

JEE Main 2012

LEVELJEE Main

A spherical balloon is filled with $4500 π$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72 π$ cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases $49$ minutes after the leakage began is

The equation of the tangent to the curve $y = x + \frac{4}{x ^{2}}$ , that is parallel to the $x$ -axis, is

(A)

y = 1

(B)

y = 2

(C)

y = 3

(D)

y = 0

JEE Main 2009

LEVELJEE Main

The shortest distance between the line $y - x = 1$ and the curve $x = y^{2}$ is

(A)

\frac{2 3}{8}

(B)

\frac{3 2}{5}

(C)

\frac{3}{4}

(D)

\frac{3 2}{8}

JEE Main 2006

LEVELJEE Main

A body falling from rest under gravity passes a certain point P. It was at a distance of 400 m from P, 4s prior to passing through P. If $g = 10 m/s^{2}$ , then the height above the point P from where the body began to fall is

Angle between the tangents to the curve $y = x^{2} - 5 x + 6$ at the points $(2, 0)$ and $(3, 0)$ is

(A)

π

(B)

\frac{π}{2}

(C)

\frac{π}{6}

(D)

\frac{π}{4}

JEE Main 2005

LEVELJEE Main

The normal to the curve $x = a (cos θ + θ sin θ)$ , $y = a (sin θ - θ cos θ)$ at any point ' $θ$ ' is such that

(A)

it passes through the origin

(B)

it makes an angle

\frac{π}{2} + θ

with the

x

-axis

(C)

it passes through

(a \frac{π}{2}, - a)

(D)

it is at a constant distance from the origin

JEE Main 2005

LEVELJEE Main

A spherical iron ball $10$ cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of $50$ cm $^{3}$ /min. When the thickness of ice is $5$ cm, then the rate at which the thickness of ice decreases is

(A)

\frac{1}{36 π} cm/min

(B)

\frac{1}{18 π} cm/min

(C)

\frac{1}{54 π} cm/min

(D)

\frac{5}{6 π} cm/min

JEE Main 2005

LEVELJEE Main

Two points A and B move from rest along a straight line with constant acceleration f and f' respectively. If A takes m sec. more than B and describes n units more than B in acquiring the same speed then

(A)

(f - f^{'}) m^{2} = f f^{'} n

(B)

(f + f^{'}) m^{2} = f f^{'} n

(C)

\frac{1}{2} (f + f^{'}) m^{2} = f f^{'} n^{2}

(D)

(f^{'} - f) n = \frac{1}{2} f f^{'} m^{2}

JEE Main 2004

LEVELJEE Main

A point on the parabola $y^{2} = 18 x$ at which the ordinate increases at twice the rate of the abscissa is

The normal to the curve $x = a (1 + cos θ)$ , $y = a sin θ$ at ' $θ$ ' always passes through the fixed point

The tangent to the curve $y = e^{x}$ drawn at the point $(c, e^{c})$ intersects the line joining the points $(c - 1, e^{c - 1})$ and $(c + 1, e^{c + 1})$

(A)

on the left of

x = c

(B)

on the right of

x = c

The point(s) on the curve $y^{3} + 3 x^{2} = 12 y$ where the tangent is vertical, is (are)

(A)

(\pm \frac{4}{3}, - 2)

(B)

(\pm \frac{11}{3}, 1)

(C)

(0, 0)

(D)

(\pm \frac{4}{3}, 2)

JEE Advanced 2001

LEVELJEE Main

The triangle formed by the tangent to the curve $f (x) = x^{2} + b x - b$ at the point $(1, 1)$ and the coordinate axes, lies in the first quadrant. If its area is $2$ , then the value of $b$ is

(A)

- 1

(B)

3

(C)

- 3

(D)

1

JEE Advanced 2000

LEVELBoard

If the normal to the curve $y = f (x)$ at the point $(3, 4)$ makes an angle $\frac{3 π}{4}$ with the positive $x$ -axis, then $f^{'} (3) =$

(A)

- 1

(B)

- \frac{3}{4}

(C)

\frac{4}{3}

(D)

1

JEE Advanced 1986

LEVELJEE Main

If the line $a x + b y + c = 0$ is a normal to the curve $x y = 1$ , then

* Multiple Correct Options

(A)

a > 0, b > 0

(B)

a > 0, b < 0

(C)

a < 0, b > 0

(D)

a < 0, b < 0

The normal to the curve $x = a (cos θ + θ sin θ)$ , $y = a (sin θ - θ cos θ)$ at any point ' $θ$ ' is such that

(A)

it makes a constant angle with the

x

-axis

(B)

it passes through the origin

(C)

it is at a constant distance from the origin

Tangent at a point $P_{1}$ {other than $(0, 0)$ } on the curve $y = x^{3}$ meets the curve again at $P_{2}$ . The tangent at $P_{2}$ meets the curve at $P_{3}$ , and so on. Show that the abscissae of $P_{1}, P_{2}, P_{3}, \dots, P_{n}$ form a GP. Also find the ratio $[area (△ P_{1} P_{2} P_{3})] / [area (△ P_{2} P_{3} P_{4})]$ .

JEE Advanced 2005

LEVELJEE Main

If $∣ f (x_{1}) - f (x_{2}) ∣ < (x_{1} - x_{2})^{2}$ , for all $x_{1}, x_{2} \in R$ . Find the equation of tangent to the curve $y = f (x)$ at the point $(1, 2)$ .

JEE Advanced 2003

LEVELJEE Main

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

JEE Advanced 2003

LEVELJEE Advanced

A right circular cone with radius R and height H contains a liquid which evaporates at a rate proportional to its surface area in contact with air (proportionality constant = k > 0). Find the time after which the cone is empty.

JEE Advanced 1994

LEVELJEE Main

Let $C$ be the curve $y^{3} - 3 x y + 2 = 0$ . If $H$ is the set of points on the curve $C$ where the tangent is horizontal and $V$ is the set of the point on the curve $C$ where the tangent is vertical then $H = .........$ and $V = .........$

JEE Advanced 1994

LEVELJEE Main

The curve $y = a x^{3} + b x^{2} + c x + 5$ , touches the $x$ -axis at $P (- 2, 0)$ and cuts the $y$ axis at a point $Q$ , where its gradient is 3. Find $a, b, c$ .

JEE Advanced 1993

LEVELJEE Main

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Consider: f(x)=tan−1(1−sinx1+sinx​​),x∈(0,2π​). A normal to y=f(x) at x=6π​ also passes through the point

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The normal to the curve, x2+2xy−3y2=0, at (1,1)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation of the tangent to the curve y=x+x24​, that is parallel to the x-axis, is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The shortest distance between the line y−x=1 and the curve x=y2 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Angle between the tangents to the curve y=x2−5x+6 at the points (2,0) and (3,0) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The normal to the curve x=a(cosθ+θsinθ), y=a(sinθ−θcosθ) at any point 'θ' is such that

Two points A and B move from rest along a straight line with constant acceleration f and f' respectively. If A takes m sec. more than B and describes n units more than B in acquiring the same speed then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A point on the parabola y2=18x at which the ordinate increases at twice the rate of the abscissa is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The normal to the curve x=a(1+cosθ), y=asinθ at 'θ' always passes through the fixed point

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The tangent to the curve y=ex drawn at the point (c,ec) intersects the line joining the points (c−1,ec−1) and (c+1,ec+1)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The point(s) on the curve y3+3x2=12y where the tangent is vertical, is (are)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the normal to the curve y=f(x) at the point (3,4) makes an angle 43π​ with the positive x-axis, then f′(3)=

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the line ax+by+c=0 is a normal to the curve xy=1, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The normal to the curve x=a(cosθ+θsinθ), y=a(sinθ−θcosθ) at any point 'θ' is such that

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If ∣f(x1​)−f(x2​)∣<(x1​−x2​)2, for all x1​,x2​∈R. Find the equation of tangent to the curve y=f(x) at the point (1,2).

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find a point on the curve x2+2y2=6 whose distance from the line x+y=7 is minimum.

A right circular cone with radius R and height H contains a liquid which evaporates at a rate proportional to its surface area in contact with air (proportionality constant = k > 0). Find the time after which the cone is empty.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The curve y=ax3+bx2+cx+5, touches the x-axis at P(−2,0) and cuts the y axis at a point Q, where its gradient is 3. Find a,b,c.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Find the equation of the normal to the curve y=(1+x)y+sin−1(sin2x) at x=0

Consider: $f (x) = tan^{- 1} (\frac{1 + s i n x}{1 - s i n x}), x \in (0, \frac{π}{2})$ . A normal to $y = f (x)$ at $x = \frac{π}{6}$ also passes through the point

The normal to the curve, $x^{2} + 2 x y - 3 y^{2} = 0$ , at $(1, 1)$

The equation of the tangent to the curve $y = x + \frac{4}{x ^{2}}$ , that is parallel to the $x$ -axis, is

The shortest distance between the line $y - x = 1$ and the curve $x = y^{2}$ is

Angle between the tangents to the curve $y = x^{2} - 5 x + 6$ at the points $(2, 0)$ and $(3, 0)$ is

The normal to the curve $x = a (cos θ + θ sin θ)$ , $y = a (sin θ - θ cos θ)$ at any point ' $θ$ ' is such that

A point on the parabola $y^{2} = 18 x$ at which the ordinate increases at twice the rate of the abscissa is

The normal to the curve $x = a (1 + cos θ)$ , $y = a sin θ$ at ' $θ$ ' always passes through the fixed point

The tangent to the curve $y = e^{x}$ drawn at the point $(c, e^{c})$ intersects the line joining the points $(c - 1, e^{c - 1})$ and $(c + 1, e^{c + 1})$

The point(s) on the curve $y^{3} + 3 x^{2} = 12 y$ where the tangent is vertical, is (are)

If the normal to the curve $y = f (x)$ at the point $(3, 4)$ makes an angle $\frac{3 π}{4}$ with the positive $x$ -axis, then $f^{'} (3) =$

If the line $a x + b y + c = 0$ is a normal to the curve $x y = 1$ , then

The normal to the curve $x = a (cos θ + θ sin θ)$ , $y = a (sin θ - θ cos θ)$ at any point ' $θ$ ' is such that

If $∣ f (x_{1}) - f (x_{2}) ∣ < (x_{1} - x_{2})^{2}$ , for all $x_{1}, x_{2} \in R$ . Find the equation of tangent to the curve $y = f (x)$ at the point $(1, 2)$ .

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

The curve $y = a x^{3} + b x^{2} + c x + 5$ , touches the $x$ -axis at $P (- 2, 0)$ and cuts the $y$ axis at a point $Q$ , where its gradient is 3. Find $a, b, c$ .

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