If a hyperbola passes through the point $\mathrm{P}(10,16)$ and it has vertices at $(\pm 6,0),$ then the equation of the normal to it at $P$ is

The vertices of a hyperbola are at $(0, 0)$ and $(10, 0)$ and one of its foci is at $(18, 0)$. The equation of the hyperbola is

Let $0 < \theta  < \frac{\pi }{2}$. If the eccentricity of the hyperbola $\frac{{{x^2}}}{{{{\cos }^2}\,\theta }} - \frac{{{y^2}}}{{{{\sin }^2}\,\theta }} = 1$ is greater than $2$, then the length of its latus rectum lies in the interval

If the foci of the ellipse $\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{{b^2}}} = 1$ coincide with the foci of the hyperbola $\frac{{{x^2}}}{{144}} - \frac{{{y^2}}}{{81}} = \frac{1}{{25}},$ then $b^2$ is equal to

If the eccentricity of a hyperbola $\frac{{{x^2}}}{9} - \frac{{{y^2}}}{{{b^2}}} = 1,$ which passes through $(K, 2),$ is $\frac{{\sqrt {13} }}{3},$ then the value of $K^2$ is

A hyperbola has its centre at the origin, passes through the point $(4, 2)$ and has transverse axis of length $4$ along the $x -$ axis. Then the eccentricity of the hyperbola is