The eccentricity of the hyperbola $16x^{2} - 3y^{2} - 32x - 12y - 44 = 0$ is

If 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 $\frac{36}{13}$ and its eccentricity is $\frac{\sqrt{13}}{3}$,then $a - b =$

If the eccentricity of a hyperbola is $\sqrt{3}$,then the eccentricity of its conjugate hyperbola is:

The area (in sq units) of the triangle formed by the tangent at $(\sqrt{3}, 0)$ to the hyperbola $x^2-3y^2=3$ with the pair of asymptotes of the hyperbola is

Let $LL^{\prime}$ be the latus rectum through the focus $S$ of a hyperbola and $A^{\prime}$ be the opposite vertex of the hyperbola. If triangle $A^{\prime}LL^{\prime}$ is equilateral,then the eccentricity of the hyperbola is

If the tangent and normal to a rectangular hyperbola $xy = c^2$ at a variable point cut off intercepts $a_1, a_2$ on the $x$-axis and $b_1, b_2$ on the $y$-axis,then $(a_1a_2 + b_1b_2)$ is