The length of the latus rectum of the hyperbola $9x^2 - 16y^2 + 72x - 32y - 16 = 0$ is

Let $P (10, 2 \sqrt{15})$ be a point on the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ whose foci are $S$ and $S'$. If the length of its latus rectum is $8$,then the square of the area of $\Delta PSS'$ is equal to:

If the centre,vertex,and focus of a hyperbola are $(0, 0)$,$(4, 0)$,and $(6, 0)$ respectively,then the equation of the hyperbola is

Let $H: \frac{-x^2}{a^2}+\frac{y^2}{b^2}=1$ be the hyperbola,whose eccentricity is $\sqrt{3}$ and the length of the latus rectum is $4\sqrt{3}$. Suppose the point $(\alpha, 6)$,where $\alpha > 0$,lies on $H$. If $\beta$ is the product of the focal distances of the point $(\alpha, 6)$,then $\alpha^2+\beta$ is equal to:

The equation of the hyperbola referred to the coordinate axes as axes of symmetry,whose distance between the foci is $16$ and eccentricity is $\sqrt{2}$,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 $k$.