(D) For ellipse $E$, eccentricity $e = \frac{\sqrt{3}}{2}$ and directrix $x = \frac{a}{e} = \frac{4\sqrt{6}}{3}$.Thus, $a = \frac{4\sqrt{6}}{3} \cdot

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(D) For ellipse $E$, eccentricity $e = \frac{\sqrt{3}}{2}$ and directrix $x = \frac{a}{e} = \frac{4\sqrt{6}}{3}$.Thus, $a = \frac{4\sqrt{6}}{3} \cdot \frac{\sqrt{3}}{2} = 2\sqrt{2}$.Since $b^2 = a^2(1 - e^2)$, we have $b^2 = 8(1 - 3/4) = 2$, so $b = \sqrt{2}$.For hyperbola $H$, eccentricity $e_H = a = 2\sqrt{2}$.The length of the latus rectum of $H$ is $\frac{2b_H^2}{a_H} = 2b = 2\sqrt{2}$.Also, for hyperbola $H$, $b_H^2 = a_H^2(e_H^2 - 1) = a_H^2(8 - 1) = 7a_H^2$.Substituting this into the latus rectum equation: $\frac{2(7a_H^2)}{a_H} = 2\sqrt{2} \implies 14a_H = 2\sqrt{2} \implies a_H = \frac{\sqrt{2}}{7}$.The distance between the foci of $H$ is $2a_H e_H = 2 \cdot \frac{\sqrt{2}}{7} \cdot 2\sqrt{2} = \frac{8}{7}$.

Class 11 Mathematics 10-2. Parabola, Ellipse, Hyperbola Mix Examples-Conic Sections

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The eccentricity of an ellipse $E$ with centre at the origin $O$ is $\frac{\sqrt{3}}{2}$ and its directrices are $x = \pm \frac{4\sqrt{6}}{3}$. Let $H : \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ be a hyperbola whose eccentricity is equal to the length of semi-major axis of $E$, and whose length of latus rectum is equal to the length of minor axis of $E$. Then the distance between the foci of $H$ is :

JEE MAIN 2026 All JEE MAIN

A
$\frac{4\sqrt{2}}{\sqrt{7}}$
B
$\frac{4\sqrt{2}}{7}$
C
$\frac{4}{\sqrt{7}}$
D
$\frac{8}{7}$

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10-2. Parabola, Ellipse, Hyperbola

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Mix Examples-Conic Sections

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JEE MAIN 2026 Paper All JEE MAIN years →

Match the conics in Column-$I$ with the statements/expressions in Column-$II$.

Column-$I$ Column-$II$

$A$. Circle $P$. Locus of point $(h, k)$ such that the line $hx + ky = 1$ touches the circle $x^2 + y^2 = 4$

$B$. Parabola $Q$. Point $z$ in the complex plane satisfies $|z + 2| - |z - 2| = \pm 3$

$C$. Hyperbola $R$. Eccentricity of the conic lies in the interval $[1, \infty)$

$S$. Point $z$ in the complex plane satisfies $Re(z + 1)^2 = |z|^2 + 1$

Column-$I$	Column-$II$
$A$. Circle	$P$. Locus of point $(h, k)$ such that the line $hx + ky = 1$ touches the circle $x^2 + y^2 = 4$
$B$. Parabola	$Q$. Point $z$ in the complex plane satisfies $\|z + 2\| - \|z - 2\| = \pm 3$
$C$. Hyperbola	$R$. Eccentricity of the conic lies in the interval $[1, \infty)$
	$S$. Point $z$ in the complex plane satisfies $Re(z + 1)^2 = \|z\|^2 + 1$

Difficult

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The graph of the conic $x^2-(y-1)^2=1$ has one tangent line with positive slope that passes through the origin. If the point of tangency is $(a, b)$,then find the value of $\sin^{-1}\left(\frac{a}{b}\right)$.

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Let $e_1$ and $e_2$ be the eccentricities of the ellipse $\frac{x^2}{b^2} + \frac{y^2}{25} = 1$ and the hyperbola $\frac{x^2}{16} - \frac{y^2}{b^2} = 1$,respectively. If $b < 5$ and $e_1 e_2 = 1$,then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is:

DifficultJEE MAIN 2025

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If the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and the hyperbola $\frac{x^2}{4}-\frac{y^2}{b^2}=1$ coincide,then $b^2$ is equal to

MediumAP EAMCET 2013

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Let the eccentricity of an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ be the reciprocal of the eccentricity of the hyperbola $2x^2 - 2y^2 = 1$. If the ellipse intersects the hyperbola at right angles,then the square of the length of the latus-rectum of the ellipse is $................$.

DifficultJEE MAIN 2023

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The graph of the conic $x^2-(y-1)^2=1$ has one tangent line with positive slope that passes through the origin. If the point of tangency is $(a, b)$,then find the value of $\sin^{-1}\left(\frac{a}{b}\right)$.

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