The equation of the tangents to the hyperbola $4x^2 -y^2 = 12$ are $y = 4x+ c_1 \,$$ \& \, y = 4x + c_2,$ then $|c_1 -c_2|$ is equal to –

If $P$ is a point on the hyperbola $16{x^2} – 9{y^2} = 144$ whose foci are ${S_1}$ and ${S_2}$, then $P{S_1}- P{S_2} = $

The length of the latus rectum and directrices of a hyperbola with eccentricity e are 9 and $\mathrm{x}= \pm \frac{4}{\sqrt{3}}$, respectively. Let the line $y-\sqrt{3} \mathrm{x}+\sqrt{3}=0$ touch this hyperbola at $\left(\mathrm{x}_0, \mathrm{y}_0\right)$. If $\mathrm{m}$ is the product of the focal distances of the point $\left(\mathrm{x}_0, \mathrm{y}_0\right)$, then $4 \mathrm{e}^2+\mathrm{m}$ is equal to ………..

If the eccentricity of the standard hyperbola passing, through the point $(4, 6)$ is $2$, then the equation of the tangent to the hyperbola at $(4, 6)$ is

Find the equation of axis of the given hyperbola $\frac{{{x^2}}}{3} – \frac{{{y^2}}}{2} = 1$ which is equally inclined to the axes