The eccentricity of the hyperbola $5{x^2} - 4{y^2} + 20x + 8y = 4$ 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

The locus of a point $P\left( {\alpha ,\beta } \right)$ moving under the condition that the line $y = \alpha x + \beta $ is a tangent to the hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ is

If the tangents drawn to the hyperbola $4y^2 = x^2 + 1$ intersect the co-ordinate axes at the distinct points $A$ and $B$, then the locus of the mid point of $AB$ is

Let a line $L: 2 x+y=k, k\,>\,0$ be a tangent to the hyperbola $x^{2}-y^{2}=3 .$ If $L$ is also a tangent to the parabola $y^{2}=\alpha x$, then $\alpha$ is equal to :

Let $H _{ n }=\frac{ x ^2}{1+ n }-\frac{ y ^2}{3+ n }=1, n \in N$. Let $k$ be the smallest even value of $n$ such that the eccentricity of $H _{ k }$ is a rational number. If $l$ is length of the latus return of $H _{ k }$, then $21 l$ is equal to $.......$.