The line $y = mx + c$ touches the curve $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,if

For the hyperbola $\frac{x^{2}}{\cos^{2} \alpha} - \frac{y^{2}}{\sin^{2} \alpha} = 1$, which of the following remains fixed when $\alpha$ varies?

The normal to the rectangular hyperbola $xy = c^2$ at the point $t_1$ meets the curve again at the point $t_2$. Then the value of $t_1^3 t_2$ is

Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be $2a$ and $2b$,respectively,and one focus and the corresponding directrix of this hyperbola be $(-5, 0)$ and $5x + 9 = 0$,respectively. If the product of the focal distances of a point $(\alpha, 2\sqrt{5})$ on the hyperbola is $p$,then $4p$ is equal to . . . . . . .

$P$ is a point on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. $N$ is the foot of the perpendicular from $P$ on the transverse axis. The tangent to the hyperbola at $P$ meets the transverse axis at $T$. If $O$ is the centre of the hyperbola,then $OT \cdot ON$ is equal to:

The locus of the point of intersection of the straight lines $\frac{x}{a} - \frac{y}{b} = m$ and $\frac{x}{a} + \frac{y}{b} = \frac{1}{m}$ is