Equation of Tangent and Normal

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JEE Main 2016

Let $P$ be the point on the parabola, $y^{2} = 8 x$ which is at a minimum distance from the centre $C$ of the circle, $x^{2} + (y + 6)^{2} = 1$ . Then the equation of the circle, passing through $C$ and having its centre at $P$ is:

x^{2} + y^{2} + \frac{x}{4} + 2 y - 24 = 0

(B)

x^{2} + y^{2} - 4 x + 9 y + 18 = 0

(C)

x^{2} + y^{2} - 4 x + 8 y + 12 = 0

(D)

x^{2} + y^{2} - x + 4 y - 12 = 0

JEE Main 2015

LEVELJEE Advanced

The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse $\frac{x ^{2}}{9} + \frac{y ^{2}}{5} = 1$ , is:

(A)

\frac{27}{2}

(B)

(C)

\frac{27}{4}

(D)

JEE Main 2014

LEVELJEE Main

The locus of the foot of perpendicular drawn from the centre of the ellipse $x^{2} + 3 y^{2} = 6$ on any tangent to it is

(A)

(x^{2} + y^{2})^{2} = 6 x^{2} + 2 y^{2}

(B)

(x^{2} + y^{2})^{2} = 6 x^{2} - 2 y^{2}

(C)

(x^{2} - y^{2})^{2} = 6 x^{2} + 2 y^{2}

(D)

(x^{2} - y^{2})^{2} = 6 x^{2} - 2 y^{2}

JEE Main 2014

LEVELJEE Main

The slope of the line touching both the parabolas $y^{2} = 4 x$ and $x^{2} = - 32 y$ is

(A)

\frac{1}{8}

(B)

\frac{2}{3}

(C)

\frac{1}{2}

(D)

\frac{3}{2}

JEE Main 2012

LEVELJEE Advanced

Statement-1 : An equation of a common tangent to the parabola $y^{2} = 163 x$ and the ellipse $2 x^{2} + y^{2} = 4$ is $y = 2 x + 23$ Statement-2 : If the line $y = m x + \frac{4 3}{m}, (m \neq = 0)$ is a common tangent to the parabola $y^{2} = 163 x$ and the ellipse $2 x^{2} + y^{2} = 4$ , then $m$ satisfies $m^{4} + 2 m^{2} = 24$

(A)

Statement-1 is false, Statement-2 is true.

(B)

Statement-1 is true, statement-2 is true; statement-2 is a correct explanation for Statement-1.

(C)

Statement-1 is true, statement-2 is true; statement-2 is not a correct explanation for Statement-1.

(D)

Statement-1 is true, statement-2 is false.

JEE Main 2011

LEVELJEE Main

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

3 /4

JEE Main 2010

LEVELJEE Main

If two tangents drawn from a point $P$ to the parabola $y^{2} = 4 x$ are at right angles, then the locus of $P$ is

(A)

2 x + 1 = 0

(B)

x = - 1

(C)

2 x - 1 = 0

(D)

x = 1

JEE Main 2007

LEVELJEE Main

The equation of a tangent to the parabola $y^{2} = 8 x$ is $y = x + 2$ . The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is

(A)

(2, 4)

(B)

(- 2, 0)

(C)

(- 1, 1)

(D)

(0, 2)

JEE Main 2005

LEVELJEE Main

The locus of a point $P (α, β)$ moving under the condition that the line $y = α x + β$ is a tangent to the hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ is

The normal at the point $(b t_{1}^{2}, 2 b t_{1})$ on a parabola meets the parabola again in the point $(b t_{2}^{2}, 2 b t_{2})$ , then

(A)

t_{2} = t_{1} + \frac{2}{t _{1}}

(B)

t_{2} = - t_{1} - \frac{2}{t _{1}}

(C)

t_{2} = - t_{1} + \frac{2}{t _{1}}

(D)

t_{2} = t_{1} - \frac{2}{t _{1}}

JEE Advanced 2016

LEVELJEE Advanced

Let $P$ be the point on the parabola $y^{2} = 4 x$ which is at the shortest distance from the center $S$ of the circle $x^{2} + y^{2} - 4 x - 16 y + 64 = 0$ . Let $Q$ be the point on the circle dividing the line segment $S P$ internally. Then

* Multiple Correct Options

(A)

S P = 25

(B)

S Q : QP = (5 + 1) : 2

(C)

the

x

-intercept of the normal to the parabola at

P

is 6

(D)

the slope of the tangent to the circle at

Q

\frac{1}{2}

JEE Advanced 2015

LEVELJEE Advanced

Let $E_{1}$ and $E_{2}$ be two ellipses whose centers are at the origin. The major axes of $E_{1}$ and $E_{2}$ lie along the $x$ -axis and the $y$ -axis, respectively. Let $S$ be the circle $x^{2} + (y - 1)^{2} = 2$ . The straight line $x + y = 3$ touches the curves $S, E_{1}$ and $E_{2}$ at $P, Q$ and $R$ respectively. Suppose that $P Q = P R = \frac{2 2}{3}$ . If $e_{1}$ and $e_{2}$ are the eccentricities of $E_{1}$ and $E_{2}$ , respectively, then the correct expression(s) is (are)

* Multiple Correct Options

(A)

e_{1}^{2} + e_{2}^{2} = \frac{43}{40}

(B)

e_{1} e_{2} = \frac{7}{2 10}

(C)

∣ e_{1}^{2} - e_{2}^{2} ∣ = \frac{5}{8}

(D)

e_{1} e_{2} = \frac{3}{4}

JEE Advanced 2015

LEVELJEE Advanced

Consider the hyperbola $H : x^{2} - y^{2} = 1$ and a circle $S$ with center $N (x_{2}, 0)$ . Suppose that $H$ and $S$ touch each other at a point $P (x_{1}, y_{1})$ with $x_{1} > 1$ and $y_{1} > 0$ . The common tangent to $H$ and $S$ at $P$ intersects the $x$ -axis at point $M$ . If $(l, m)$ is the centroid of the triangle $Δ P M N$ , then the correct expression(s) is(are)

* Multiple Correct Options

(A)

\frac{d l}{d x _{1}} = 1 - \frac{1}{3 x _{1}^{2}}

for

x_{1} > 1

(B)

\frac{d m}{d x _{1}} = \frac{x _{1}}{3 x _{1}^{2} - 1}

for

x_{1} > 1

(C)

\frac{d l}{d x _{1}} = 1 + \frac{1}{3 x _{1}^{2}}

for

x_{1} > 1

(D)

\frac{d m}{d y _{1}} = \frac{1}{3}

for

y_{1} > 0

JEE Advanced 2012

LEVELJEE Main

Tangents are drawn to the hyperbola $\frac{x ^{2}}{9} - \frac{y ^{2}}{4} = 1$ , parallel to the straight line $2 x - y = 1$ . The points of contact of the tangents on the hyperbola are

* Multiple Correct Options

(A)

(\frac{9}{2 2}, \frac{1}{2})

(B)

(- \frac{9}{2 2}, - \frac{1}{2})

(C)

(33, - 22)

(D)

(- 33, 22)

JEE Advanced 2011

LEVELJEE Main

Let $P (6, 3)$ be a point on the hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ . If the normal at the point $P$ intersects the $x$ -axis at $(9, 0)$ , then the eccentricity of the hyperbola is

(A)

5/2

(B)

3/2

(C)

2

(D)

3

JEE Advanced 2011

LEVELJEE Main

Let $L$ be a normal to the parabola $y^{2} = 4 x$ . If $L$ passes through the point $(9, 6)$ , then $L$ is given by

* Multiple Correct Options

(A)

y - x + 3 = 0

(B)

y + 3 x - 33 = 0

(C)

y + x - 15 = 0

(D)

y - 2 x + 12 = 0

JEE Advanced 2009

LEVELJEE Advanced

The normal at a point $P$ on the ellipse $x^{2} + 4 y^{2} = 16$ meets the $x$ -axis at $Q$ . If $M$ is the mid point of the line segment $P Q$ , then the locus of $M$ intersects the latus rectums of the given ellipse at the points

(A)

(\pm \frac{3 5}{2}, \pm \frac{2}{7})

(B)

(\pm \frac{3 5}{2}, \pm \frac{19}{4})

(C)

(\pm 23, \pm \frac{1}{7})

(D)

(\pm 23, \pm \frac{4 3}{7})

JEE Advanced 2009

LEVELJEE Advanced

The tangent $P T$ and the normal $P N$ to the parabola $y^{2} = 4 a x$ at a point $P$ on it meet its axis at points $T$ and $N$ , respectively. The locus of the centroid of the triangle $P T N$ is a parabola whose

* Multiple Correct Options

(A)

vertex is

(\frac{2 a}{3}, 0)

(B)

directrix is

x = 0

(C)

latus rectum is

\frac{2 a}{3}

(D)

focus is

(a, 0)

JEE Advanced 2006

LEVELJEE Main

The equations of the common tangents to the parabola $y = x^{2}$ and $y = - (x - 2)^{2}$ is/are

* Multiple Correct Options

(A)

y = 4 (x - 1)

(B)

y = 0

(C)

y = - 4 (x - 1)

(D)

y = - 30 x - 50

JEE Advanced 2005

LEVELJEE Main

The minimum area of triangle formed by the tangent to the $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ & coordinate axes is

(A)

ab

sq. units

(B)

\frac{a ^{2} + b ^{2}}{2}

sq. units

(C)

\frac{( a + b ) ^{2}}{2}

sq. units

(D)

\frac{a ^{2} + ab + b ^{2}}{3}

sq. units

JEE Advanced 2004

LEVELJEE Main

If tangents are drawn to the ellipse $x^{2} + 2 y^{2} = 2$ , then the locus of the mid-point of the intercept made by the tangents between the coordinate axes is

(A)

\frac{1}{2 x ^{2}} + \frac{1}{4 y ^{2}} = 1

(B)

\frac{1}{4 x ^{2}} + \frac{1}{2 y ^{2}} = 1

(C)

\frac{x ^{2}}{2} + \frac{y ^{2}}{4} = 1

(D)

\frac{x ^{2}}{4} + \frac{y ^{2}}{2} = 1

JEE Advanced 2004

LEVELJEE Main

The angle between the tangents drawn from the point $(1, 4)$ to the parabola $y^{2} = 4 x$ is

(A)

π /6

(B)

π /4

(C)

π /3

(D)

π /2

JEE Advanced 2004

LEVELJEE Main

If the line $2 x + 6 y = 2$ touches the hyperbola $x^{2} - 2 y^{2} = 4$ , then the point of contact is

(A)

(- 2, 6)

(B)

(- 5, 26)

(C)

(1/2, 1/ 6)

(D)

(4, - 6)

JEE Advanced 2003

LEVELJEE Main

Tangent is drawn to ellipse $\frac{x ^{2}}{27} + y^{2} = 1$ at $(33 cos θ, sin θ)$ (where $θ \in (0, π /2)$ ). Then the value of $θ$ such that sum of intercepts on axes made by this tangent is minimum, is

(A)

π /3

(B)

π /6

(C)

π /8

(D)

π /4

JEE Advanced 2003

LEVELJEE Main

The area of the quadrilateral formed by the tangents at the end points of latus rectum to the ellipse $\frac{x ^{2}}{9} + \frac{y ^{2}}{5} = 1$ , is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse 9x2​+5y2​=1, is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The locus of the foot of perpendicular drawn from the centre of the ellipse x2+3y2=6 on any tangent to it is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The slope of the line touching both the parabolas y2=4x and x2=−32y is

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

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If two tangents drawn from a point P to the parabola y2=4x are at right angles, then the locus of P is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation of a tangent to the parabola y2=8x is y=x+2. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The locus of a point P(α,β) moving under the condition that the line y=αx+β is a tangent to the hyperbola a2x2​−b2y2​=1 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The normal at the point (bt12​,2bt1​) on a parabola meets the parabola again in the point (bt22​,2bt2​), then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Tangents are drawn to the hyperbola 9x2​−4y2​=1, parallel to the straight line 2x−y=1. The points of contact of the tangents on the hyperbola are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let P(6,3) be a point on the hyperbola a2x2​−b2y2​=1. If the normal at the point P intersects the x-axis at (9,0), then the eccentricity of the hyperbola is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let L be a normal to the parabola y2=4x. If L passes through the point (9,6), then L is given by

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equations of the common tangents to the parabola y=x2 and y=−(x−2)2 is/are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The minimum area of triangle formed by the tangent to the a2x2​+b2y2​=1 & coordinate axes is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If tangents are drawn to the ellipse x2+2y2=2, then the locus of the mid-point of the intercept made by the tangents between the coordinate axes is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The angle between the tangents drawn from the point (1,4) to the parabola y2=4x is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the line 2x+6​y=2 touches the hyperbola x2−2y2=4, then the point of contact is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The area of the quadrilateral formed by the tangents at the end points of latus rectum to the ellipse 9x2​+5y2​=1, is

The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse $\frac{x ^{2}}{9} + \frac{y ^{2}}{5} = 1$ , is:

The locus of the foot of perpendicular drawn from the centre of the ellipse $x^{2} + 3 y^{2} = 6$ on any tangent to it is

The slope of the line touching both the parabolas $y^{2} = 4 x$ and $x^{2} = - 32 y$ is

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

If two tangents drawn from a point $P$ to the parabola $y^{2} = 4 x$ are at right angles, then the locus of $P$ is

The equation of a tangent to the parabola $y^{2} = 8 x$ is $y = x + 2$ . The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is

The locus of a point $P (α, β)$ moving under the condition that the line $y = α x + β$ is a tangent to the hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ is

The normal at the point $(b t_{1}^{2}, 2 b t_{1})$ on a parabola meets the parabola again in the point $(b t_{2}^{2}, 2 b t_{2})$ , then

Tangents are drawn to the hyperbola $\frac{x ^{2}}{9} - \frac{y ^{2}}{4} = 1$ , parallel to the straight line $2 x - y = 1$ . The points of contact of the tangents on the hyperbola are

Let $P (6, 3)$ be a point on the hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ . If the normal at the point $P$ intersects the $x$ -axis at $(9, 0)$ , then the eccentricity of the hyperbola is

Let $L$ be a normal to the parabola $y^{2} = 4 x$ . If $L$ passes through the point $(9, 6)$ , then $L$ is given by

The equations of the common tangents to the parabola $y = x^{2}$ and $y = - (x - 2)^{2}$ is/are

The minimum area of triangle formed by the tangent to the $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ & coordinate axes is

If tangents are drawn to the ellipse $x^{2} + 2 y^{2} = 2$ , then the locus of the mid-point of the intercept made by the tangents between the coordinate axes is

The angle between the tangents drawn from the point $(1, 4)$ to the parabola $y^{2} = 4 x$ is

If the line $2 x + 6 y = 2$ touches the hyperbola $x^{2} - 2 y^{2} = 4$ , then the point of contact is

The area of the quadrilateral formed by the tangents at the end points of latus rectum to the ellipse $\frac{x ^{2}}{9} + \frac{y ^{2}}{5} = 1$ , is