Standard Equations of Parabola, Ellipse, and Hyperbola

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JEE Main 2013

LEVELJEE Advanced

Given : A circle, $2 x^{2} + 2 y^{2} = 5$ and a parabola, $y^{2} = 45 x$ . Statement-1 : An equation of a common tangent to these curves is $y = x + 5$ . Statement-2 : If the line, $y = m x + \frac{5}{m} (m \neq = 0)$ is their common tangent, then $m$ satisfies $m^{4} - 3 m^{2} + 2 = 0$ .

(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.

JEE Main 2012

LEVELJEE Main

An ellipse is drawn by taking a diameter of the circle $(x - 1)^{2} + y^{2} = 1$ as its semi-minor axis and a diameter of the circle $x^{2} + (y - 2)^{2} = 4$ as semi-major axis. If the centre of the ellipse is at the origin and its axes are the coordinate axes, then the equation of the ellipse is:

(A)

4 x^{2} + y^{2} = 4

(B)

x^{2} + 4 y^{2} = 8

(C)

4 x^{2} + y^{2} = 8

(D)

x^{2} + 4 y^{2} = 16

JEE Main 2011

LEVELJEE Main

Equation of the ellipse whose axes are the axes of coordinates and which passes through the point $(- 3, 1)$ and has eccentricity $\frac{2}{5}$ 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

JEE Main 2009

LEVELJEE Main

The ellipse $x^{2} + 4 y^{2} = 4$ is inscribed in a rectangle aligned with the coordinate axes, which in turn is inscribed in another ellipse that passes through the point $(4, 0)$ . Then the equation of the ellipse is:

(A)

x^{2} + 12 y^{2} = 16

(B)

4 x^{2} + 48 y^{2} = 48

(C)

4 x^{2} + 64 y^{2} = 48

(D)

x^{2} + 16 y^{2} = 16

JEE Main 2008

LEVELJEE Main

A parabola has the origin as its focus and the line $x = 2$ as the directrix. Then the vertex of the parabola is at

(A)

(0, 2)

(B)

(1, 0)

(C)

(0, 1)

(D)

(2, 0)

JEE Main 2007

LEVELJEE Main

For the Hyperbola $\frac{x ^{2}}{c o s ^{2} α} - \frac{y ^{2}}{s i n ^{2} α} = 1$ , which of the following remains constant when $α$ varies?

(A)

abscissae of vertices

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}

JEE Main 2005

LEVELJEE Main

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

JEE Main 2005

LEVELJEE Main

A circle touches the x- axis and also touches the circle with centre at $(0, 3)$ and radius 2. The locus of the centre of the circle is

If $a \neq = 0$ and the line $2 b x + 3 cy + 4 d = 0$ passes through the points of intersection of the parabolas $y^{2} = 4 a x$ and $x^{2} = 4 a y$ , then

(A)

d^{2} + (3 b - 2 c)^{2} = 0

(B)

d^{2} + (3 b + 2 c)^{2} = 0

(C)

d^{2} + (2 b - 3 c)^{2} = 0

(D)

d^{2} + (2 b + 3 c)^{2} = 0

JEE Main 2004

LEVELJEE Main

The eccentricity of an ellipse, with its centre at the origin, is $1/2$ . If one of the directrices is $x = 4$ , then the equation of the ellipse is:

(A)

4 x^{2} + 3 y^{2} = 1

(B)

3 x^{2} + 4 y^{2} = 12

(C)

4 x^{2} + 3 y^{2} = 12

(D)

3 x^{2} + 4 y^{2} = 1

JEE Main 2003

LEVELJEE Main

Two stones are projected from the top of a cliff h metres high, with the same speed u, so as to hit the ground at the same spot. If one of the stones is projected horizontally and the other is projected at an angle $θ$ to the horizontal then $tan θ$ equals

(A)

u \frac{2}{g h}

(B)

\frac{2 u}{g h}

(C)

2 g \frac{u}{h}

(D)

2 h \frac{u}{g}

JEE Main 2003

LEVELJEE Main

The foci of the ellipse $\frac{x ^{2}}{16} + \frac{y ^{2}}{b ^{2}} = 1$ and the hyperbola $\frac{x ^{2}}{144} - \frac{y ^{2}}{81} = \frac{1}{25}$ coincide. Then the value of $b^{2}$ is

Two common tangents to the circle $x^{2} + y^{2} = 2 a^{2}$ and parabola $y^{2} = 8 a x$ are

(A)

x = \pm (y + 2 a)

(B)

y = \pm (x + 2 a)

(C)

x = \pm (y + a)

(D)

y = \pm (x + a)

JEE Advanced 2015

LEVELJEE Advanced

Let $P$ and $Q$ be distinct points on the parabola $y^{2} = 2 x$ such that a circle with $P Q$ as diameter passes through the vertex $O$ of the parabola. If $P$ lies in the first quadrant and the area of the triangle $Δ O P Q$ is $32$ , then which of the following is (are) the coordinates of $P$ ?

* Multiple Correct Options

(A)

(4, 22)

(B)

(9, 32)

(C)

(\frac{1}{4}, \frac{1}{2})

(D)

(1, 2)

JEE Advanced 2014

LEVELJEE Advanced

The common tangents to the circle $x^{2} + y^{2} = 2$ and the parabola $y^{2} = 8 x$ touch the circle at the points $P, Q$ and the parabola at the points $R, S$ . Then the area of the quadrilateral $P QR S$ is

The ellipse $E_{1} : \frac{x ^{2}}{9} + \frac{y ^{2}}{4} = 1$ is inscribed in a rectangle $R$ whose sides are parallel to the coordinate axes. Another ellipse $E_{2}$ passing through the point $(0, 4)$ circumscribes the rectangle $R$ . The eccentricity of the ellipse $E_{2}$ is

(A)

\frac{2}{2}

(B)

\frac{3}{2}

(C)

\frac{1}{2}

(D)

\frac{3}{4}

JEE Advanced 2011

LEVELJEE Main

Let $(x, y)$ be any point on the parabola $y^{2} = 4 x$ . Let $P$ be the point that divides the line segment from $(0, 0)$ to $(x, y)$ in the ratio $1 : 3$ . Then the locus of $P$ is

(A)

x^{2} = y

(B)

y^{2} = 2 x

(C)

y^{2} = x

(D)

x^{2} = 2 y

JEE Advanced 2010

LEVELJEE Advanced

Let $A$ and $B$ be two distinct points on the parabola $y^{2} = 4 x$ . If the axis of the parabola touches a circle of radius $r$ having $A B$ as its diameter, then the slope of the line joining $A$ and $B$ can be

* Multiple Correct Options

(A)

\frac{1}{r}

(B)

- \frac{1}{r}

(C)

\frac{2}{r}

(D)

- \frac{2}{r}

JEE Advanced 2009

LEVELJEE Main

The line passing through the extremity $A$ of the major axis and extremity $B$ of the minor axis of the ellipse $x^{2} + 9 y^{2} = 9$ meets its auxiliary circle at the point $M$ . Then the area of the triangle with vertices at $A, M$ and the origin $O$ is

(A)

31/10

(B)

29/10

(C)

21/10

(D)

27/10

JEE Advanced 2008

LEVELJEE Main

Consider the two curves $C_{1} : y^{2} = 4 x, C_{2} : x^{2} + y^{2} - 6 x + 1 = 0$ . Then,

(A)

C_{1}

and

C_{2}

touch each other only at one point

(B)

C_{1}

and

C_{2}

touch each other exactly at two points

(C)

C_{1}

and

C_{2}

intersect (but do not touch) at exactly two points

(D)

C_{1}

and

C_{2}

neither intersect nor touch each other

JEE Advanced 2008

LEVELJEE Main

Consider a branch of the hyperbola $x^{2} - 2 y^{2} - 22 x - 42 y - 6 = 0$ with vertex at the point $A$ . Let $B$ be one of the end points of its latus rectum. If $C$ is the focus of the hyperbola nearest to the point $A$ , then the area of the triangle $A B C$ is

(A)

1 - \frac{2}{3}

(B)

\frac{3}{2} - 1

(C)

1 + \frac{2}{3}

(D)

\frac{3}{2} + 1

JEE Advanced 2008

LEVELJEE Main

Let $P (x_{1}, y_{1})$ and $Q (x_{2}, y_{2}), y_{1} < 0, y_{2} < 0$ , be the end points of the latus rectum of the ellipse $x^{2} + 4 y^{2} = 4$ . The equations of parabolas with latus rectum $P Q$ are

* Multiple Correct Options

(A)

x^{2} + 23 y = 3 + 3

(B)

x^{2} - 23 y = 3 + 3

(C)

x^{2} + 23 y = 3 - 3

(D)

x^{2} - 23 y = 3 - 3

JEE Advanced 2007

LEVELJEE Main

A hyperbola, having the transverse axis of length $2 sin θ$ , is confocal with the ellipse $3 x^{2} + 4 y^{2} = 12$ . Then its equation is

(A)

x^{2} cosec^{2} θ - y^{2} sec^{2} θ = 1

(B)

x^{2} sec^{2} θ - y^{2} cosec^{2} θ = 1

(C)

x^{2} sin^{2} θ - y^{2} cos^{2} θ = 1

(D)

x^{2} cos^{2} θ - y^{2} sin^{2} θ = 1

JEE Advanced 2007

LEVELBoard

STATEMENT-1 : The curve $y = \frac{- x ^{2}}{2} + x + 1$ is symmetric with respect to the line $x = 1$ . because STATEMENT-2 : A parabola is symmetric about its axis.

(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.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Equation of the ellipse whose axes are the axes of coordinates and which passes through the point (−3,1) and has eccentricity 52​​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A parabola has the origin as its focus and the line x=2 as the directrix. Then the vertex of the parabola is at

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For the Hyperbola cos2αx2​−sin2αy2​=1, which of the following remains constant when α varies?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The locus of the vertices of the family of parabolas y=3a3x2​+2a2x​−2a is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let P be the point (1,0) and Q a point on the locus y2=8x. The locus of mid point of PQ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A circle touches the x- axis and also touches the circle with centre at (0,3) and radius 2. The locus of the centre of the circle is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If a=0 and the line 2bx+3cy+4d=0 passes through the points of intersection of the parabolas y2=4ax and x2=4ay, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The eccentricity of an ellipse, with its centre at the origin, is 1/2. If one of the directrices is x=4, then the equation of the ellipse is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The foci of the ellipse 16x2​+b2y2​=1 and the hyperbola 144x2​−81y2​=251​ coincide. Then the value of b2 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Two common tangents to the circle x2+y2=2a2 and parabola y2=8ax are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let (x,y) be any point on the parabola y2=4x. Let P be the point that divides the line segment from (0,0) to (x,y) in the ratio 1:3. Then the locus of P is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Consider the two curves C1​:y2=4x,C2​:x2+y2−6x+1=0. Then,

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let P(x1​,y1​) and Q(x2​,y2​),y1​<0,y2​<0, be the end points of the latus rectum of the ellipse x2+4y2=4. The equations of parabolas with latus rectum PQ are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A hyperbola, having the transverse axis of length 2sinθ, is confocal with the ellipse 3x2+4y2=12. Then its equation is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } STATEMENT-1 : The curve y=2−x2​+x+1 is symmetric with respect to the line x=1. because STATEMENT-2 : A parabola is symmetric about its axis.

Equation of the ellipse whose axes are the axes of coordinates and which passes through the point $(- 3, 1)$ and has eccentricity $\frac{2}{5}$ is

A parabola has the origin as its focus and the line $x = 2$ as the directrix. Then the vertex of the parabola is at

For the Hyperbola $\frac{x ^{2}}{c o s ^{2} α} - \frac{y ^{2}}{s i n ^{2} α} = 1$ , which of the following remains constant when $α$ varies?

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

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 circle touches the x- axis and also touches the circle with centre at $(0, 3)$ and radius 2. The locus of the centre of the circle is

If $a \neq = 0$ and the line $2 b x + 3 cy + 4 d = 0$ passes through the points of intersection of the parabolas $y^{2} = 4 a x$ and $x^{2} = 4 a y$ , then

The eccentricity of an ellipse, with its centre at the origin, is $1/2$ . If one of the directrices is $x = 4$ , then the equation of the ellipse is:

The foci of the ellipse $\frac{x ^{2}}{16} + \frac{y ^{2}}{b ^{2}} = 1$ and the hyperbola $\frac{x ^{2}}{144} - \frac{y ^{2}}{81} = \frac{1}{25}$ coincide. Then the value of $b^{2}$ is

Two common tangents to the circle $x^{2} + y^{2} = 2 a^{2}$ and parabola $y^{2} = 8 a x$ are

Let $(x, y)$ be any point on the parabola $y^{2} = 4 x$ . Let $P$ be the point that divides the line segment from $(0, 0)$ to $(x, y)$ in the ratio $1 : 3$ . Then the locus of $P$ is

Consider the two curves $C_{1} : y^{2} = 4 x, C_{2} : x^{2} + y^{2} - 6 x + 1 = 0$ . Then,

Let $P (x_{1}, y_{1})$ and $Q (x_{2}, y_{2}), y_{1} < 0, y_{2} < 0$ , be the end points of the latus rectum of the ellipse $x^{2} + 4 y^{2} = 4$ . The equations of parabolas with latus rectum $P Q$ are

A hyperbola, having the transverse axis of length $2 sin θ$ , is confocal with the ellipse $3 x^{2} + 4 y^{2} = 12$ . Then its equation is

STATEMENT-1 : The curve $y = \frac{- x ^{2}}{2} + x + 1$ is symmetric with respect to the line $x = 1$ . because STATEMENT-2 : A parabola is symmetric about its axis.