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 $4(2 \sqrt{2}+\sqrt{14})$. If the eccentricity $H$ is $\frac{\sqrt{11}}{2}$, then value of $a^{2}+b^{2}$ is equal to

If angle between asymptotes of hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{3} = 4$ is $\frac{\pi }{3}$, then its conjugate hyperbola is

The vertices of a hyperbola $H$ are $(\pm 6,0)$ 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 $\sqrt{2} x + y =2 \sqrt{2}$. If $d$ is the length of the line segment of $N$ between $H$ and the $y$-axis then $d ^2$ is equal to $............$.

Length of latusrectum of curve $xy = 7x + 5y$ is

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 $S$ be the focus of the hyperbola $\frac{x^2}{3}-\frac{y^2}{5}=1$, on the positive $\mathrm{x}$-axis. Let $\mathrm{C}$ be the circle with its centre at $\mathrm{A}(\sqrt{6}, \sqrt{5})$ and passing through the point $\mathrm{S}$. if $\mathrm{O}$ is the origin and $\mathrm{SAB}$ is a diameter of $\mathrm{C}$ then the square of the area of the triangle $OSB$ is equal to ....................