The locus of the foot of the perpendicular from the centre of the hyperbola $xy = c^2$  on a variable tangent is :

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 $…………$.

Latus rectum of the conic satisfying the differential equation, $ x dy + y dx = 0$  and passing through the point $ (2, 8) $ is :

Let tangents drawn from point $C(0,-b)$ to hyperbola $\frac{{{x^2}}}{{{a^2}}} – \frac{{{y^2}}}{{{b^2}}} = 1$ touches hyperbola at points $A$ and $B.$ If $\Delta ABC$ is a right angled triangle, then $\frac{a^2}{b^2}$ is equal to –

If $PQ$ is a double ordinate of the hyperbola $\frac{{{x^2}}}{{{a^2}}} – \frac{{{y^2}}}{{{b^2}}} = 1$ such that $OPQ$ is an equilateral triangle, $O$ being the center of the hyperbola. then the $'e'$ eccentricity of the hyperbola, satisfies