A hyperbola passes through the points $(3, 2)$ and $(-17, 12)$ and has its centre at origin and transverse axis is along $x$ – axis. The length of its transverse axis is

If $(4, 0)$ and $(-4, 0)$ be the vertices and $(6, 0)$ and $(-6, 0)$ be the foci of a hyperbola, then its eccentricity is

Let $H : \frac{ x ^{2}}{ a ^{2}}-\frac{y^{2}}{ b ^{2}}=1$, a $>0, b >0$, be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is $4(2 \sqrt{2}+\sqrt{14})$. If the eccentricity $H$ is $\frac{\sqrt{11}}{2}$, then value of $a^{2}+b^{2}$ is equal to

The locus of a point $P(\alpha ,\,\beta )$ 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

A common tangent $T$ to the curves $C_{1}: \frac{x^{2}}{4}+\frac{y^{2}}{9}=1$ and $C_{2}: \frac{x^{2}}{42}-\frac{y^{2}}{143}=1$ does not pass through the fourth quadrant. If $T$ touches $C _{1}$ at ( $\left.x _{1}, y _{1}\right)$ and $C _{2}$ at $\left( x _{2}, y _{2}\right)$, then $\left|2 x _{1}+ x _{2}\right|$ is equal to $……$