The equation of the common tangent to the curves $y^2 = 8x$ and $xy = -1$ 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

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

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

The one which does not represent a hyperbola is

The eccentricity of the hyperbola ${x^2} - {y^2} = 25$ is