The equation of the chord of the hyperbola $25x^{2} - 16y^{2} = 400$ whose midpoint is $(5, 3)$ is:

For all real values of $m$,the straight line $y = mx + \sqrt{9m^2 - 4}$ is a tangent to the curve:

If $P(\frac{\pi}{4})$ and $Q(\frac{3 \pi}{4})$ are two points on the hyperbola $4 x^2-y^2-8 x-2 y-13=0$ in parametric form,then the distance between $P$ and $Q$ is

If the normals drawn to the hyperbola $xy=4$ at $(\alpha_i, \beta_i)$ for $i=1, 2, 3, 4$ are concurrent at the point $(a, b)$,then $\frac{(\alpha_1+\alpha_2+\alpha_3+\alpha_4)}{(\beta_1+\beta_2+\beta_3+\beta_4)}(\alpha_1 \alpha_2 \alpha_3 \alpha_4) =$

The locus of the point of intersection of the tangents at the endpoints of normal chords of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is

If $e_{1}$ and $e_{2}$ represent the eccentricity of the curves $16x^{2}-9y^{2}=144$ and $9x^{2}-16y^{2}=144$ respectively,then $\frac{1}{e_{1}^{2}}+\frac{1}{e_{2}^{2}}$ is equal to