If $\theta$ is the acute angle between the tangents drawn from the point $(2,3)$ to the hyperbola $5x^2-6y^2-30=0$,then $\tan \theta=$

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 $a - b =$

The coordinates of the foci of the rectangular hyperbola $xy = c^2$ are

Let the origin be the centre,$(\pm 3, 0)$ be the foci,and $\frac{3}{2}$ be the eccentricity of a hyperbola. Then the line $2x - y - 1 = 0$

Let a line $L_{1}$ be tangent to the hyperbola $\frac{x^{2}}{16}-\frac{y^{2}}{4}=1$ and let $L_{2}$ be the line passing through the origin and perpendicular to $L_{1}$. If the locus of the point of intersection of $L_{1}$ and $L_{2}$ is $(x^{2}+y^{2})^{2} = \alpha x^{2}+\beta y^{2}$,then $\alpha+\beta$ is equal to

The centre of the hyperbola $9x^{2} - 36x - 16y^{2} + 96y - 252 = 0$ is: