Let the foci of a hyperbola $\mathrm{H}$ coincide with the foci of the ellipse $E: \frac{(x-1)^2}{100}+\frac{(y-1)^2}{75}=1$ and the eccentricity of the hyperbola $\mathrm{H}$ be the reciprocal of the eccentricity of the ellipse $E$. If the length of the transverse axis of $\mathrm{H}$ is $\alpha$ and the length of its conjugate axis is $\beta$, then $3 \alpha^2+2 \beta^2$ is equal to :

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

Point from which two distinct tangents can be drawn on two different branches of the hyperbola $\frac{{{x^2}}}{{25}} - \frac{{{y^2}}}{{16}} = \,1$ but no two different tangent can be drawn to the circle $x^2 + y^2 = 36$ is

For the Hyperbola ${x^2}{\sec ^2}\theta - {y^2}cose{c^2}\theta = 1$ which of the following remains constant when $\theta $ varies $= ?$

What will be equation of that chord of hyperbola $25{x^2} - 16{y^2} = 400$, whose mid point is $(5, 3)$

The normal to the rectangular hyperbola $xy = c^2$ at the point $'t_1'$ meets the curve again at the point $'t_2'$ . Then the value of $t_{1}^{3} t_{2}$ is