The length of the latus rectum of $x^2+3y^2=12$ is

How many real tangents can be drawn to the ellipse $5x^2 + 9y^2 = 32$ from the point $(2, 3)$?

If $F_1$ and $F_2$ are the feet of the perpendiculars from the foci $S_1$ and $S_2$ of an ellipse $\frac{x^2}{5} + \frac{y^2}{3} = 1$ on the tangent at any point $P$ on the ellipse,then $(S_1 F_1) (S_2 F_2)$ is equal to

The locus of a variable point whose distance from $(-2, 0)$ is $\frac{2}{3}$ times its distance from the line $x = -\frac{9}{2}$ is

The equation of the tangent to the ellipse $x^2+16y^2=16$ which makes an angle $60^{\circ}$ with the $X$-axis is

The equations of the directrices of the ellipse $16x^2 + 25y^2 = 400$ are