Let a line $L_{1}$ be tangent to the hyperbola $\frac{x^{2}}{16}-\frac{y^{2}}{4}=1$ and let $L_{2}$ be the line passing through the origin and perpendicular to $L _{1}$. If the locus of the point of intersection of $L_{1}$ and $L_{2}$ is $\left(x^{2}+y^{2}\right)^{2}=$ $\alpha x^{2}+\beta y^{2}$, then $\alpha+\beta$ is equal to

The equation of the hyperbola referred to its axes as axes of coordinate and whose distance between the foci is $16$ and eccentricity is $\sqrt 2 $, is

Let the eccentricity of the hyperbola $H : \frac{ x ^{2}}{ a ^{2}}-\frac{ y ^{2}}{ b ^{2}}=1$ be $\sqrt{\frac{5}{2}}$ and length of its latus rectum be $6 \sqrt{2}$, If $y =2 x + c$ is a tangent to the hyperbola $H$, then the value of $c ^{2}$ is equal to

$P$  is a point on the hyperbola $\frac{{{x^2}}}{{{a^2}}} – \frac{{{y^2}}}{{{b^2}}}$ $= 1, N $ is the foot of the perpendicular from $P$  on the transverse axis. The tangent to the hyperbola at $P$  meets the transverse axis at $ T$  . If $O$ is the centre of the hyperbola, the $OT. ON$  is equal to :

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