If $F_1$ and $F_2$ be the feet of the perpendicular 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 line, $ lx + my + n = 0$  will cut the ellipse $\frac{{{x^2}}}{{{a^2}}}$ $+$ $\frac{{{y^2}}}{{{b^2}}}$ $= 1 $ in points whose eccentric angles differ by $\pi /2$  if :

An ellipse is described by using an endless string which is passed over two pins. If the axes are $6\ cm$ and $4\ cm$, the necessary length of the string and the distance between the pins respectively in $cm$, are

The equation of the ellipse whose foci are $( \pm 5,\;0)$ and one of its directrix is $5x = 36$, is

A rod $AB$ of length $15\,cm$ rests in between two coordinate axes in such a way that the end point A lies on $x-$ axis and end point $B$ lies on $y-$ axis. A point $P(x,\, y)$ is taken on the rod in such a way that $AP =6\, cm .$ Show that the locus of $P$ is an ellipse.

Let $\theta$ be the acute angle between the tangents to the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{1}=1$ and the circle $x^{2}+y^{2}=3$ at their point of intersection in the first quadrant. Then $\tan \theta$ is equal to :