If the straight line $x\cos \alpha + y\sin \alpha = p$ be a tangent to the hyperbola $\frac{{{x^2}}}{{{a^2}}} – \frac{{{y^2}}}{{{b^2}}} = 1$, then

For the hyperbola $H : x ^{2}- y ^{2}=1$ and the ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a>b>0$, let the

$(1)$ eccentricity of $E$ be reciprocal of the eccentricity of $H$, and

$(2)$ the line $y=\sqrt{\frac{5}{2}} x+K$ be a common tangent of $E$ and $H$ Then $4\left(a^{2}+b^{2}\right)$ is equal to

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

If the length of the transverse and conjugate axes of a hyperbola be $8$ and $6$ respectively, then the difference focal distances of any point of the hyperbola will be

Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola $49 y^{2}-16 x^{2}=784$