Foci, Directrices, and Eccentricity

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JEE Main 2016

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is :

\frac{2}{3}

(B)

3

(C)

\frac{4}{3}

(D)

\frac{4}{3}

JEE Main 2013

LEVELJEE Main

The equation of the circle passing through the foci of the ellipse $\frac{x ^{2}}{16} + \frac{y ^{2}}{9} = 1$ , and having centre at $(0, 3)$ is

(A)

x^{2} + y^{2} - 6 y - 7 = 0

(B)

x^{2} + y^{2} - 6 y + 7 = 0

(C)

x^{2} + y^{2} - 6 y - 5 = 0

(D)

x^{2} + y^{2} - 6 y + 5 = 0

JEE Main 2008

LEVELJEE Main

A focus of an ellipse is at the origin. The directrix is the line $x = 4$ and the eccentricity is $\frac{1}{2}$ . Then the length of the semi-major axis is

(A)

\frac{8}{3}

(B)

\frac{2}{3}

(C)

\frac{4}{3}

(D)

\frac{5}{3}

JEE Main 2006

LEVELBoard

In an ellipse, the distance between its foci is 6 and minor axis is 8. Then its eccentricity is

(A)

\frac{3}{5}

(B)

\frac{1}{2}

(C)

\frac{4}{5}

(D)

\frac{1}{5}

JEE Main 2005

LEVELJEE Main

An ellipse has $O B$ as semi minor axis, $F$ and $F^{'}$ its focii and the angle $F B F^{'}$ is a right angle. Then the eccentricity of the ellipse is

(A)

\frac{1}{2}

(B)

\frac{1}{2}

(C)

\frac{1}{4}

(D)

\frac{1}{3}

JEE Advanced 2011

LEVELJEE Main

Let the eccentricity of the hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ be a reciprocal to that of the ellipse $x^{2} + 4 y^{2} = 4$ . If the hyperbola passes through a focus of the ellipse, then

* Multiple Correct Options

(A)

the equation of the hyperbola is

\frac{x ^{2}}{3} - \frac{y ^{2}}{2} = 1

(B)

a focus of the hyperbola is

(2, 0)

(C)

the eccentricity of the hyperbola is

\frac{5}{3}

(D)

the equation of the hyperbola is

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

JEE Advanced 2009

LEVELJEE Advanced

An ellipse intersects the hyperbola $2 x^{2} - 2 y^{2} = 1$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then

* Multiple Correct Options

(A)

equation of ellipse is

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

(B)

the foci of ellipse are

(\pm 1, 0)

(C)

equation of ellipse is

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

(D)

the foci of ellipse are

(\pm 2, 0)

JEE Advanced 2003

LEVELJEE Main

For hyperbola $\frac{x ^{2}}{c o s ^{2} α} - \frac{y ^{2}}{s i n ^{2} α} = 1$ which of the following remains constant with change in ' $α$ '?

(A)

abscissae of vertices

The radius of the circle passing through the foci of the ellipse $\frac{x ^{2}}{16} + \frac{y ^{2}}{9} = 1$ , and having its centre at $(0, 3)$ is

(A)

(B)

(C)

12

(D)

7/2

JEE Main 2019 (9 January)

LEVELJEE Main

Let $0 < θ < \frac{π}{2}$ . If the eccentricity of the hyperbola $\frac{x ^{2}}{c o s ^{2} θ} - \frac{y ^{2}}{s i n ^{2} θ} = 1$ is greater than 2, then the length of its latus rectum lies in the interval :

JEE Main 11 Jan 2019 (Evening)

LEVELJEE Main

If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the eccentricity of the hyperbola is:

(A)

(B)

13/8

(C)

13/12

(D)

13/6

JEE Main 2020 (9 January Shift 1)

LEVELJEE Main

If $e_{1}$ and $e_{2}$ are the eccentricities of the ellipse, $\frac{x ^{2}}{18} + \frac{y ^{2}}{4} = 1$ and the hyperbola, $\frac{x ^{2}}{9} - \frac{y ^{2}}{4} = 1$ respectively and $(e_{1}, e_{2})$ is a point on the ellipse, $15 x^{2} + 3 y^{2} = k$ . Then $k$ is equal to:

(A)

14

(B)

15

(C)

17

(D)

16

JEE Main 2020 (7 Jan Morning)

LEVELJEE Main

If distance between the foci of an ellipse is 6 and distance between its directrices is 12, then length of its latus rectum is

(A)

(B)

32

(C)

(D)

22

JEE Main 2020 (9 Jan Morning)

LEVELJEE Main

If $e_{1}$ and $e_{2}$ are eccentricities of the ellipse $\frac{x ^{2}}{18} + \frac{y ^{2}}{4} = 1$ and the hyperbola $\frac{x ^{2}}{9} - \frac{y ^{2}}{4} = 1$ respectively and if the point $(e_{1}, e_{2})$ lies on ellipse $15 x^{2} + 3 y^{2} = k$ . Then find value of $k$

JEE Main 2020 (4 Sep Morning)

LEVELJEE Main

Let $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1 (a > b)$ be a given ellipse, length of whose latus rectum is 10. If its eccentricity is the maximum value of the function, $ϕ (t) = \frac{5}{12} + t - t^{2}$ , then $a^{2} + b^{2}$ is equal to :

JEE Main 2020 (5 Sep Morning)

LEVELJEE Main

If the co-ordinates of two points $A$ and $B$ are $(7, 0)$ and $(- 7, 0)$ respectively and $P$ is any point on the conic, $9 x^{2} + 16 y^{2} = 144$ , then $P A + P B$ is equal to :

JEE Main 2021 (27 July Shift 1)

LEVELJEE Advanced

A ray of light through $(2, 1)$ is reflected at a point $P$ on the $y$ -axis and then passes through the point $(5, 3)$ . If this reflected ray is the directrix of an ellipse with eccentricity $\frac{1}{3}$ and the distance of the nearer focus from this directrix is $\frac{8}{53}$ , then the equation of the other directrix can be:

(A)

11 x + 7 y + 8 = 0

11 x + 7 y - 15 = 0

(B)

11 x - 7 y - 8 = 0

11 x + 7 y + 15 = 0

(C)

2 x - 7 y + 29 = 0

2 x - 7 y - 7 = 0

(D)

2 x - 7 y - 39 = 0

2 x - 7 y - 7 = 0

JEE Main 2022 (29 June Shift 1)

LEVELJEE Main

Let $H : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1, a > 0, b > 0$ , be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is $4 (22 + 14)$ . If the eccentricity $H$ is $\frac{11}{2}$ , then value of $a^{2} + b^{2}$ is equal to ______.

JEE Main 2022 (25 July Shift 2)

LEVELJEE Main

If the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ meets the line $\frac{x}{7} + \frac{y}{2 6} = 1$ on the x-axis and the line $\frac{x}{7} - \frac{y}{2 6} = 1$ on the y-axis, then the eccentricity of the ellipse is

(A)

\frac{5}{7}

(B)

\frac{2 6}{7}

(C)

\frac{3}{7}

(D)

\frac{2 5}{7}

JEE Main 2023 (15 Apr Shift 1)

LEVELJEE Main

Let an ellipse with centre $(1, 0)$ and latus rectum of length $\frac{1}{2}$ have its major axis along x-axis. If its minor axis subtends an angle $6 0^{\circ}$ at the foci, then the square of the sum of the lengths of its minor and major axes is equal to _______.

JEE Main 2023 (31 January Shift 2)

LEVELJEE Main

Let H be the hyperbola, whose foci are $(1 \pm 2, 0)$ and eccentricity is $2$ . Then the length of its latus rectum is

(A)

(B)

(C)

\frac{5}{2}

(D)

\frac{3}{2}

JEE Main 2024 (09 Apr Shift 1)

LEVELJEE Main

Let $f (x) = x^{2} + 9, g (x) = \frac{x}{x - 9}$ and $a = f \circ g (10), b = g \circ f (3)$ . If $e$ and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\frac{x ^{2}}{a} + \frac{y ^{2}}{b} = 1$ , then $8 e^{2} + l^{2}$ is equal to.

JEE Main 2024 (30 Jan Shift 1)

LEVELJEE Main

If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :

(A)

\frac{5}{3}

(B)

\frac{3}{2}

(C)

\frac{1}{3}

(D)

\frac{2}{5}

JEE Main 2024 (27 Jan Shift 2)

LEVELJEE Main

Let $e_{1}$ be the eccentricity of the hyperbola $\frac{x ^{2}}{16} - \frac{y ^{2}}{9} = 1$ and $e_{2}$ be the eccentricity of the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b$ , which passes through the foci of the hyperbola. If $e_{1} e_{2} = 1$ , then the length of the chord of the ellipse parallel to the x-axis and passing through $(0, 2)$ is :

(A)

45

(B)

\frac{8 5}{3}

(C)

\frac{10 5}{3}

(D)

35

JEE Main 2025 April

LEVELJEE Main

If $S$ and $S^{'}$ are the foci of the ellipse $\frac{x ^{2}}{18} + \frac{y ^{2}}{9} = 1$ and $P$ be a point on the ellipse, then $min (S P \cdot S^{'} P) + max (S P \cdot S^{'} P)$ is equal to :

(A)

3 (1 + 2)

(B)

3 (6 + 2)

(C)

(D)

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The equation of the circle passing through the foci of the ellipse 16x2​+9y2​=1, and having centre at (0,3) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A focus of an ellipse is at the origin. The directrix is the line x=4 and the eccentricity is 21​. Then the length of the semi-major axis is

In an ellipse, the distance between its foci is 6 and minor axis is 8. Then its eccentricity is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } An ellipse has OB as semi minor axis, F and F′ its focii and the angle FBF′ is a right angle. Then the eccentricity of the ellipse is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let the eccentricity of the hyperbola a2x2​−b2y2​=1 be a reciprocal to that of the ellipse x2+4y2=4. If the hyperbola passes through a focus of the ellipse, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For hyperbola cos2αx2​−sin2αy2​=1 which of the following remains constant with change in 'α'?

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The radius of the circle passing through the foci of the ellipse 16x2​+9y2​=1, and having its centre at (0,3) is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let 0<θ<2π​. If the eccentricity of the hyperbola cos2θx2​−sin2θy2​=1 is greater than 2, then the length of its latus rectum lies in the interval :

If a hyperbola has length of its conjugate axis equal to 5 and the distance between its foci is 13, then the eccentricity of the hyperbola is:

If distance between the foci of an ellipse is 6 and distance between its directrices is 12, then length of its latus rectum is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the co-ordinates of two points A and B are (7​,0) and (−7​,0) respectively and P is any point on the conic, 9x2+16y2=144, then PA+PB is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the ellipse a2x2​+b2y2​=1 meets the line 7x​+26​y​=1 on the x-axis and the line 7x​−26​y​=1 on the y-axis, then the eccentricity of the ellipse is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let H be the hyperbola, whose foci are (1±2​,0) and eccentricity is 2​. Then the length of its latus rectum is

If the length of the minor axis of ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If S and S′ are the foci of the ellipse 18x2​+9y2​=1 and P be a point on the ellipse, then min(SP⋅S′P)+max(SP⋅S′P) is equal to :

The equation of the circle passing through the foci of the ellipse $\frac{x ^{2}}{16} + \frac{y ^{2}}{9} = 1$ , and having centre at $(0, 3)$ is

A focus of an ellipse is at the origin. The directrix is the line $x = 4$ and the eccentricity is $\frac{1}{2}$ . Then the length of the semi-major axis is

An ellipse has $O B$ as semi minor axis, $F$ and $F^{'}$ its focii and the angle $F B F^{'}$ is a right angle. Then the eccentricity of the ellipse is

Let the eccentricity of the hyperbola $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ be a reciprocal to that of the ellipse $x^{2} + 4 y^{2} = 4$ . If the hyperbola passes through a focus of the ellipse, then

For hyperbola $\frac{x ^{2}}{c o s ^{2} α} - \frac{y ^{2}}{s i n ^{2} α} = 1$ which of the following remains constant with change in ' $α$ '?

The radius of the circle passing through the foci of the ellipse $\frac{x ^{2}}{16} + \frac{y ^{2}}{9} = 1$ , and having its centre at $(0, 3)$ is

Let $0 < θ < \frac{π}{2}$ . If the eccentricity of the hyperbola $\frac{x ^{2}}{c o s ^{2} θ} - \frac{y ^{2}}{s i n ^{2} θ} = 1$ is greater than 2, then the length of its latus rectum lies in the interval :

If the co-ordinates of two points $A$ and $B$ are $(7, 0)$ and $(- 7, 0)$ respectively and $P$ is any point on the conic, $9 x^{2} + 16 y^{2} = 144$ , then $P A + P B$ is equal to :

If the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ meets the line $\frac{x}{7} + \frac{y}{2 6} = 1$ on the x-axis and the line $\frac{x}{7} - \frac{y}{2 6} = 1$ on the y-axis, then the eccentricity of the ellipse is

Let H be the hyperbola, whose foci are $(1 \pm 2, 0)$ and eccentricity is $2$ . Then the length of its latus rectum is

If $S$ and $S^{'}$ are the foci of the ellipse $\frac{x ^{2}}{18} + \frac{y ^{2}}{9} = 1$ and $P$ be a point on the ellipse, then $min (S P \cdot S^{'} P) + max (S P \cdot S^{'} P)$ is equal to :