Hyperbola

The vertices of a hyperbola $H$ are $\left(\pm\,6,0\right)$ and its eccentricity is $\frac{\sqrt{5}}{2}$. Let $N$ be the normal to $H$ at a point in the first quadrant and parallel to the line $\sqrt{2}x+y=2\sqrt{2}$.…

Let $\mathrm{S}$ be the focus of the hyperbola $\frac{x^2}{3}-\frac{y^2}{5}=1$, on the positive $x$-axis. Let $\mathrm{C}$ be the circle with its centre at $A(\sqrt{6}, \sqrt{5})$ and passing through the point $S$. If…

Consider the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ having one of its focus at $\mathrm{P}(-3,0)$. If the latus ractum through its other focus subtends a right angle at P and $a^2 b^2=\alpha \sqrt{2}-\beta,…

The locus of the midpoints of the chord of the circle, $x^{2}+y^{2}=25$ which is tangent to the hyperbola, $\frac{x^{2}}{9}-\frac{y^{2}}{16}=1$ is :

If the equation of the hyperbola with foci $(4,2)$ and $(8,2)$ is $3 x^2-y^2-\alpha x+\beta y+\gamma=0$, then $\alpha+\beta+\gamma$ is equal to _____.

Let $P$ be a point on the hyperbola $H:\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$, in the first quadrant such that the area of triangle formed by $P$ and the two foci of $H$ is $2\sqrt{13}$. Then, the square of the distance of…

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…

Let the latus rectum of the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{b^{2}}=1$subtend an angle of $\frac{\pi\,}{3}$ at the centre of the hyperbola. If $b^{2}$is equal to $\frac{l}{m}(1+\sqrt{n})$, where $l$ and $m$ are…

A hyperbola passes through the foci of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$and its transverse and conjugate axes coincide with major and minor axes of the ellipse, respectively. If the product of their…

Let $R$ be a rectangle given by the lines $x=0,x=2$, $y=0$and $y=5$. Let $A\left(\alpha\,,0\right)$ and $B\left(0,\beta\,\right),\alpha\,\in\,\left[0,2\right]$ and $\beta\,\in\,\left[0,5\right]$, be such that the line…

A square $ABCD$ has all its vertices on the curve $x^{2}y^{2}=1.$The midpoints of its sides also lie on the same curve. Then, the square of area of $ABCD$ is

Let the circle $C$ touch the line $x-y+1=0$, have the centre on the positive x -axis, and cut off a chord of length $\frac{4}{\sqrt{13}}$ along the line $-3 x+2 y=1$. Let H be the hyperbola…