The point $\mathrm{P}(-2 \sqrt{6}, \sqrt{3})$ lies on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ having eccentricity $\frac{\sqrt{5}}{2} .$ If the tangent and normal at $\mathrm{P}$ to the hyperbola intersect its conjugate axis at the point $\mathrm{Q}$ and $\mathrm{R}$ respectively, then $QR$ is equal to :

If $e$ and $e’$ are the eccentricities of the ellipse $5{x^2} + 9{y^2} = 45$ and the hyperbola $5{x^2} - 4{y^2} = 45$ respectively, then $ee' = $

The tangent to the hyperbola, $x^2 - 3y^2 = 3$  at the point $\left( {\sqrt 3 \,\,,\,\,0} \right)$ when associated with two asymptotes constitutes :

Eccentricity of the hyperbola conjugate to the hyperbola $\frac{{{x^2}}}{4} - \frac{{{y^2}}}{{12}} = 1$ is

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

If ${m_1}$ and ${m_2}$are the slopes of the tangents to the hyperbola $\frac{{{x^2}}}{{25}} - \frac{{{y^2}}}{{16}} = 1$ which pass through the point $(6, 2)$, then