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 passes through the point $P\left( {\sqrt 2 ,\sqrt 3 } \right)$ has foci at $\left( { \pm 2,0} \right)$. Then the tangent to this hyperbola at  $P$ also passes through the point

The product of the perpendiculars drawn from any point on a hyperbola to its asymptotes is

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

The equation of the tangent to the hyperbola $2{x^2} - 3{y^2} = 6$ which is parallel to the line $y = 3x + 4$, is

The difference of the focal distance of any point on the hyperbola $9{x^2} - 16{y^2} = 144$, is