The locus of the point of intersection of the lines $\sqrt{3}x - y - 4\sqrt{3}k = 0$ and $\sqrt{3}kx + ky - 4\sqrt{3} = 0$ for different values of $k$ is

If a hyperbola has asymptotes $3x - 4y - 1 = 0$ and $4x - 3y - 6 = 0$,then the transverse and conjugate axes of that hyperbola are

Let $P$ be the foot of the perpendicular from the focus $S$ of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ on the line $bx-ay=0$ and let $C$ be the centre of the hyperbola. Then, the area of the rectangle whose sides are equal to $SP$ and $CP$ is

If $P(\frac{\pi}{6})$ is a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$,$S$ and $S^{\prime}$ are its foci,and $SP + S^{\prime}P - 2|SP - S^{\prime}P| = 0$,then the eccentricity $e$ is:

At which of the following points is the tangent to the hyperbola $x^2 - y^2 = 3$ parallel to the line $2x + y + 8 = 0$?

$A$ hyperbola, having the transverse axis of length $2 \sin \theta$, is confocal with the ellipse $3 x^{2}+4 y^{2}=12$. Its equation is