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 $\tan \theta=$

For different values of $\alpha$, the locus of the point of intersection of the two straight lines $\sqrt{3} x - y - 4 \sqrt{3} \alpha = 0$ and $\sqrt{3} \alpha x + \alpha y - 4 \sqrt{3} = 0$ is

If the latus rectum of a hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ subtends an angle of $60^{\circ}$ at the other focus,then the eccentricity of the hyperbola is

If $(0, \pm 4)$ and $(0, \pm 2)$ are the foci and vertices of a hyperbola,respectively,then its equation is:

Let $L(ae, b^2/a)$ be the end of the latus rectum of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ lying in the first quadrant,and let $S(ae, 0)$ be the focus of the given hyperbola. Given $L$ is $(x_1, 4)$ and $S$ is $(8, y_1)$,find the length of its transverse axis.

If the tangent drawn at a point $P(t)$ on the hyperbola $x^2-y^2=c^2$ cuts the $X$-axis at $T$ and the normal drawn at the same point $P$ cuts the $Y$-axis at $N$,then the equation of the locus of the midpoint of $TN$ is