The foci of the hyperbola $2{x^2} - 3{y^2} = 5$, is

The value of $m$, for which the line $y = mx + \frac{{25\sqrt 3 }}{3}$, is a normal to the conic $\frac{{{x^2}}}{{16}} - \frac{{{y^2}}}{9} = 1$, is

The length of the latus rectum and directrices of a hyperbola with eccentricity e are 9 and $\mathrm{x}= \pm \frac{4}{\sqrt{3}}$, respectively. Let the line $y-\sqrt{3} \mathrm{x}+\sqrt{3}=0$ touch this hyperbola at $\left(\mathrm{x}_0, \mathrm{y}_0\right)$. If $\mathrm{m}$ is the product of the focal distances of the point $\left(\mathrm{x}_0, \mathrm{y}_0\right)$, then $4 \mathrm{e}^2+\mathrm{m}$ is equal to ...........

For hyperbola $\frac{{{x^2}}}{{{{\cos }^2}\alpha }} - \frac{{{y^2}}}{{{{\sin }^2}\alpha }} = 1$ which of the following remain constant if $\alpha$ varies

Let $\mathrm{A}\,(\sec \theta, 2 \tan \theta)$ and $\mathrm{B}\,(\sec \phi, 2 \tan \phi)$, where $\theta+\phi=\pi / 2$, be two points on the hyperbola $2 \mathrm{x}^{2}-\mathrm{y}^{2}=2$. If $(\alpha, \beta)$ is the point of the intersection of the normals to the hyperbola at $\mathrm{A}$ and $\mathrm{B}$, then $(2 \beta)^{2}$ is equal to ..... .

The minimum value of ${\left( {{x_1} - {x_2}} \right)^2} + {\left( {\sqrt {2 - x_1^2}  - \frac{9}{{{x_2}}}} \right)^2}$ where ${x_1} \in \left( {0,\sqrt 2 } \right)$ and ${x_2} \in {R^ + }$.