Embibe Experts Solutions for Chapter: Hyperbola, Exercise 1: JEE Main - 10th January 2019 Shift 1

If the line $x-1=0$, is a directrix of the hyperbola $k{x}^2-y^2=6$, then the hyperbola passes through the point

An ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ passes through the vertices of the hyperbola $H:\frac{x^2}{49}-\frac{y^2}{64}=-1$. Let the major and minor axes of the ellipse $E$ coincide with the transverse and conjugate axes of the hyperbola $H$. Let the product of the eccentricities of $E$ and $H$ be $\frac{1}{2}$. If $l$ is the length of the latus rectum of the ellipse $E$, then the value of $113l$ is equal to _______.

For the hyperbola $H:x^2-y^2=1$ and the ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a>b>0$, let the

(1) eccentricity of $E$ be reciprocal of the eccentricity of $H$, and

(2) the line $y=\sqrt{\frac{5}{2}}x+K$ be a common tangent of $E$ and $H$.

Then $4\left(a^2+b^2\right)$ is equal to

Let the hyperbola $H:\frac{x^2}{a^2}-\frac{y^2}{ b^2}=1$ pass through the point $\left(2\sqrt{2},-2\sqrt{2}\right)$. 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 focal chord of the parabola $P:y^2=4x$ along the line $L:y=mx+c,m>0$ meet the parabola at the points $M$ and $N$. Let the line $L$ be a tangent to the hyperbola $H:x^2-y^2=4$. If $O$ is the vertex of $P$ and $F$ is the focus of $H$ on the positive $x$-axis, then the area of the quadrilateral $OMFN$ is

The vertices of a hyperbola $\mathrm{H}$ are $\left(\pm 6,0\right)$ and its eccentricity is $\frac{\sqrt{5}}{2}$. Let $\mathrm{N}$ be the normal to $\mathrm{H}$ at a point in the first quadrant and parallel to the line $\sqrt{2}x+y=2\sqrt{2}$. If $d$ is the length of the line segment of $\mathrm{N}$ between $\mathrm{H}$ and the $y$-axis then $d^2$ is equal to _____ .

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

Let $P\left(x_0,y_0\right)$ be the point on the hyperbola $3x^2-4y^2=36$, which is nearest to the line $3x+2y=1$. Then $\sqrt{2}\left(y_0-x_0\right)$ is equal to :