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} = $

Let $P (3\, sec\,\theta , 2\, tan\,\theta )$ and $Q\, (3\, sec\,\phi , 2\, tan\,\phi )$ where $\theta + \phi \, = \frac{\pi}{2}$ , be two distinct points on the hyperbola $\frac{{{x^2}}}{9} - \frac{{{y^2}}}{4} = 1$ . Then the ordinate of the point of intersection of the normals at $P$ and $Q$ is

The equation of the tangent to the conic ${x^2} - {y^2} - 8x + 2y + 11 = 0$ at $(2, 1)$ is

Let $\mathrm{A}\,(\sec \theta, 2 \tan \theta)$ and $\mathrm{B}\,(\sec \phi, 2 \tan \phi)$, where $\theta+\phi=\pi / 2$, be two points on the hyperbola $2 \mathrm{x}^{2}-\mathrm{y}^{2}=2$. If $(\alpha, \beta)$ is the point of the intersection of the normals to the hyperbola at $\mathrm{A}$ and $\mathrm{B}$, then $(2 \beta)^{2}$ is equal to ..... .

What will be equation of that chord of hyperbola $25{x^2} - 16{y^2} = 400$, whose mid point is $(5, 3)$

The equation of the transverse and conjugate axis of the hyperbola $16{x^2} - {y^2} + 64x + 4y + 44 = 0$ are