If $3x + 4y = 12\sqrt{2}$ is a tangent to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{9} = 1$ for some $a \in R$,then the distance between the foci of the ellipse is:

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 major and minor axes of an ellipse are along the $X$-axis and $Y$-axis respectively. If its latus rectum is of length $4$ and the distance between the foci is $4 \sqrt{2}$,then the equation of that ellipse is

The equations of the latus rectum of the ellipse $9x^2+4y^2-18x-8y-23=0$ are

$A$ variable tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ makes intercepts on both the axes. The locus of the midpoint of the portion of the tangent between the coordinate axes is

The lengths of the sides of the rectangle of greatest area that can be inscribed in the ellipse $x^2+4y^2=64$ are