Properties of Ellipse

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 a point $P$ on the ellipse, then $\left(S_1{M}_1\right)\left(S_2{M}_2\right)$ is equal to

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

A tangent at any point $\mathrm{P}$ on the ellipse $\frac{{\mathrm{x}}^2}{16}+\frac{{\mathrm{y}}^2}{8}=1$ intersect the major axis at point $\mathrm{R}$ and $\mathrm{M}, {\mathrm{M}}^{\text{'}}$ are foot of perpendiculars drawn from the focus $\mathrm{S}$ and ${\mathrm{S}}^{\text{'}}$ of the ellipse if $\mathrm{RS}: {\mathrm{RS}}^{\text{'}}=2: 3.$ Then $\mathrm{SM}$ is equal to

A tangent at any point $P$ on the ellipse $\frac{x^2}{16}+\frac{y^2}{8}=1$ intersect the major axis at point $R$ and $M,M^{\text{'}}$ are foot of perpendiculars drawn from the focus $S$ and $S^{\text{'}}$ of the ellipse. If $RS:R{S}^{\text{'}}=2:3$, then $SM$ is equal to

If a line $x+y-2=0$ is tangent to an ellipse at a point $\left(\alpha ,\beta \right)$ with foci $S_1\left(3,2\right)$ and $S_2\left(4,7\right)$, then the value of $\alpha +\beta$ is

The locus of feet of perpendicular from either foci of the ellipse $(\mathrm{x}-\mathrm{y}+1{)}^2+(2\mathrm{x}+2\mathrm{y}-6{)}^2=20$  on any tangent is:

An equilateral triangle is inscribed in an ellipse whose equation is $x^2+4y^2=4$, one vertex of triangle is $(0,1)$ and if one altitude is contained in $y$-axis and length of each side is $\sqrt{\frac{p}{q}}$  (where $p$ and $q$ are relatively prime), then the value of $\frac{p-3q}{9}=$

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 $\frac{k_1+k_2}{220}$.

Which of the following is compound?

Identify correct statement(s) about conic $\sqrt{{\left(\mathrm{x}–5\right)}^2+{\left(\mathrm{y}–7\right)}^2}+\mathrm{ }\sqrt{{\left(\mathrm{x}+1\right)}^2+{\left(\mathrm{y}+1\right)}^2}=12$

An ellipse has $OB$as semi minor axis, $F$ and $F$ it's foci and the angle $FBF\prime$ is a right angle. Then, the eccentricity of the ellipse is

A focus of an ellipse is at the origin. The directrix is the line $x=4$ and the eccentricity is $\frac{1}{2,}$ then length of semi major axis is

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

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 a point $P$ on the ellipse, then $\left(S_1{M}_1\right)\left(S_2{M}_2\right)$ is equal to

An ellipse slides between two lines at right angle to one another. The locus of its centre is