The locus of the point of intersection of the lines $\sqrt{3}x - y - 4\sqrt{3}t = 0$ and $\sqrt{3}tx + ty - 4\sqrt{3} = 0$ (where $t$ is a parameter) is a hyperbola whose eccentricity is

The directrix of the hyperbola $\frac{x^2}{9} - \frac{y^2}{4} = 1$ is:

If the eccentricity of the standard hyperbola passing through the point $(4, 6)$ is $2$,then the equation of the tangent to the hyperbola at $(4, 6)$ is

For the hyperbola $\frac{x^{2}}{\cos^{2} \alpha} - \frac{y^{2}}{\sin^{2} \alpha} = 1$, which of the following remains fixed when $\alpha$ varies?

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 line $x-1=0$ is a directrix of the hyperbola $kx^{2}-y^{2}=6$,then the hyperbola passes through which of the following points?