Let the hyperbola $H : \frac{ x ^{2}}{ a ^{2}}-\frac{ y ^{2}}{ b ^{2}}=1$ pass through the point $(2 \sqrt{2},-2 \sqrt{2})$. A parabola is drawn whose focus is same as the focus of $H$ with positive abscissa and the directrix of the parabola passes through the other focus of $H$. If the length of the latus rectum of the parabola is e times the length of the latus rectum of $H$, where $e$ is the eccentricity of $H$, then which of the following points lies on the parabola?
Let the hyperbola $H : \frac{ x ^{2}}{ a ^{2}}-\frac{ y ^{2}}{ b ^{2}}=1$ pass through the point $(2 \sqrt{2},-2 \sqrt{2})$. A parabola is drawn whose focus is same as the focus of $H$ with positive abscissa and the directrix of the parabola passes through the other focus of $H$. If the length of the latus rectum of the parabola is e times the length of the latus rectum of $H$, where $e$ is the eccentricity of $H$, then which of the following points lies on the parabola?
- [JEE MAIN 2022]
- A
$(2 \sqrt{3}, 3 \sqrt{2})$
- B
$(3 \sqrt{3},-6 \sqrt{2})$
- C
$(\sqrt{3},-\sqrt{6})$
- D
$(3 \sqrt{6}, 6 \sqrt{2})$
Similar Questions
Let $H_1: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ and $H_2:-\frac{x^2}{A^2}+\frac{y^2}{B^2}=1$ be two hyperbolas having length of latus rectums $15 \sqrt{2}$ and $12 \sqrt{5}$ respectively. Let their eccentricities be $e_1=\sqrt{\frac{5}{2}}$ and $e_2$ respectively. If the product of the lengths of their transverse axes is $100 \sqrt{10}$, then $25 e _2^2$ is equal to _____
- [JEE MAIN 2025]
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If $(4, 0)$ and $(-4, 0)$ be the vertices and $(6, 0)$ and $(-6, 0)$ be the foci of a hyperbola, then its eccentricity is
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A hyperbola passes through the points $(3, 2)$ and $(-17, 12)$ and has its centre at origin and transverse axis is along $x$ - axis. The length of its transverse axis is
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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
$(A)$ 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 $( \pm \sqrt{2}, 0)$
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
$(A)$ 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 $( \pm \sqrt{2}, 0)$
- [IIT 2009]