If the eccentricities of the hyperbolas $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ and $\frac{{{y^2}}}{{{b^2}}} - \frac{{{x^2}}}{{{a^2}}} = 1$ be e and ${e_1}$, then $\frac{1}{{{e^2}}} + \frac{1}{{e_1^2}} = $

If for a hyperbola the ratio of length of conjugate Axis to the length of transverse  axis is $3 : 2$ then the ratio of distance between the focii to the distance between the two directrices is

Equation of the normal to the hyperbola $\frac{{{x^2}}}{{25}} - \frac{{{y^2}}}{{16}} = 1$ perpendicular to the line $2x + y = 1$ is

Let the tangent drawn to the parabola $y ^{2}=24 x$ at the point $(\alpha, \beta)$ is perpendicular to the line $2 x$ $+2 y=5$. Then the normal to the hyperbola $\frac{x^{2}}{\alpha^{2}}-\frac{y^{2}}{\beta^{2}}=1$ at the point $(\alpha+4, \beta+4)$ does $NOT$ pass through the point.

If angle between asymptotes of hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{3} = 4$ is $\frac{\pi }{3}$, then its conjugate hyperbola is

The equation of the tangent parallel to $y - x + 5 = 0$ drawn to $\frac{{{x^2}}}{3} - \frac{{{y^2}}}{2} = 1$ is