Properties of Tangents and Normal of an Ellipse

If $F_1$ and $F_2$ are the feet of perpendiculars from foci $S_1$ and $S_2$ of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ on the tangent at any point $P$ on the ellipse, then the least value of $\frac{S_1{F}_1+S_2{F}_2}{2}$ is

$y=x$ is tangent to an ellipse having foci at $\left(2,3\right)$ and $\left(5,7\right)$ then which of the following is/are true

If $F_1$ and $F_2$ are the foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, then the product of perpendicular from $F_1$ and $F_2$ on the tangent drawn at any point of the ellipse, is

A light ray emanating from the point $\left(-14\sqrt{6},0\right)$ is incident on the ellipse $25x^2+49y^2=1225$ at point $Q$ with abscicca zero, then the equation of the reflected ray after first reflection is

If $F_1$ & $F_2$ are the foci of ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, the product of perpendicular from $F_1$ & $F_2$ on the tangent at point $P\left(3\cos \theta ,2\sin \theta \right)$ is

Product of the length of perpendiculars drawn from the foci upon any tangent to the ellipse $3x^2+4y^2=12$ is

The angle between the pair of tangents drawn to the ellipse $3x^2+2y^2=5$ from the point $\left(1, 2\right)$ is

If the locus of reflection of the ellipse $\frac{{\left(x-4\right)}^2}{16}+\frac{{\left(y-3\right)}^2}{9}=1$ about the line $x-y-2=0$ is $16x^2+9y^2+k_{1}x-36y+k_2=0,$ then, find $\frac{k_1+k_2}{220}$.

The area of the quadrilateral formed by the tangents at the end points of latus-rectum to the ellipse $\frac{x^2}{9}+\frac{y^2}{5}=1$ is

If $M_1$ and $M_2$ are the feet of the perpendiculars from the foci $S_1$ and $S_2$ of the ellipse $\frac{x^2}{9}+\frac{y^2}{16}=1$ on the tangent at any point $P$ on the ellipse, then $\left(S_1{M}_1\right)\left(S_2{M}_2\right)$ is equal to

A line $2x+3y=4$ is reflected from the inner surface of ellipse $\frac{x^2}{16}+\frac{y^2}{12}=1$. The reflected line will pass through