The foci of a hyperbola are $(\pm 2, 0)$ and its eccentricity is $\frac{3}{2}$. $A$ tangent,perpendicular to the line $2x + 3y = 6$,is drawn at a point in the first quadrant on the hyperbola. If the intercepts made by the tangent on the $x$- and $y$-axes are $a$ and $b$ respectively,then $|6a| + |5b|$ is equal to $..........$.

If the line $lx + my = 1$ is a normal to the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$,then $\frac{a^2}{l^2} - \frac{b^2}{m^2}$ is equal to

The tangent to the hyperbola $xy = c^2$ at the point $P(ct, c/t)$ intersects the $x$-axis at $T$ and the $y$-axis at $T'$. The normal to the hyperbola at $P$ intersects the $x$-axis at $N$ and the $y$-axis at $N'$. If the areas of the triangles $PNT$ and $PN'T'$ are $\Delta$ and $\Delta'$ respectively,then $\frac{1}{\Delta} + \frac{1}{\Delta'}$ is:

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

What is the eccentricity of the conjugate hyperbola of the hyperbola $x^2 - 3y^2 = 1$?

$A$ common tangent to $9x^{2}-16y^{2}=144$ and $x^{2}+y^{2}=9$ is