If $c$ is a real number and $\frac{x^2}{c-12}+\frac{y^2}{7-c}=1$ represents a hyperbola,then

The locus of a point of intersection of two lines $x \sqrt{3}-y=k \sqrt{3}$ and $\sqrt{3} k x+k y=\sqrt{3}$,where $k \in R$,describes:

The locus of the point of intersection of the lines $bxt - ayt = ab$ and $bx + ay = abt$ is

The equation of the directrices of the conic $x^2 + 2x - y^2 + 5 = 0$ are

$P$ 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 product of the perpendicular distances from any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ to its asymptotes is