Let the eccentricity of the hyperbola $H : \frac{ x ^{2}}{ a ^{2}}-\frac{ y ^{2}}{ b ^{2}}=1$ be $\sqrt{\frac{5}{2}}$ and length of its latus rectum be $6 \sqrt{2}$, If $y =2 x + c$ is a tangent to the hyperbola $H$, then the value of $c ^{2}$ is equal to

Let $m_1$ and $m_2$ be the slopes of the tangents drawn from the point $P (4,1)$ to the hyperbola $H: \frac{y^2}{25}-\frac{x^2}{16}=1$. If $Q$ is the point from which the tangents drawn to $H$ have slopes $\left| m _1\right|$ and $\left| m _2\right|$ and they make positive intercepts $\alpha$ and $\beta$ on the $x$ axis, then $\frac{(P Q)^2}{\alpha \beta}$ is equal to $…………$.

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 $…………$.

What is the slope of the tangent line drawn to the hyperbola $xy = a\,(a \ne 0)$ at the point $(a, 1)$

The chord $ PQ $ of the rectangular hyperbola $xy = a^2$  meets the axis of $x$ at $A ; C $ is the mid point of $  PQ\ \& 'O' $ is the origin. Then the $ \Delta ACO$  is :