G Tewani Solutions for Chapter: Ellipse, Exercise 3: DPP

An ellipse has the point $\left(1,-1\right)$ and $\left(2,-1\right)$ as its foci and $x+y=5$ as one of its tangent, then the value of $a^2+b^2$, where $a, b$ are the length of semi-major and semi-minor axis of ellipse respectively, is

An ellipse has foci at $\left(9, 20\right)$ and $\left(49, 55\right)$ in the $xy$-plane and tangent to it is the $x$-axis. The length of its major axis is

The maximum distance of the centre of the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ from the chord of contact of mutually perpendicular tangents of the ellipse is

 $P_1$  and  $P_2$ are lengths of the perpendiculars from the foci on the tangent to the ellipse and $P_3$ and $P_4$ are lengths of the perpendiculars from extremities of major axis and length of the perpendicular $P$ from the centre of the ellipse on the same tangent, then $\frac{P_1{P}_2-P^2}{P_3{P}_4-P^2}$ equals (where $e$ is the eccentricity of the ellipse)

From the focus $\left(-5, 0\right)$ of the ellipse $\frac{x^2}{45}+\frac{y^2}{20}=1$, a ray of light is sent which makes an angle ${\cos }^{-1}\left(\frac{-1}{\sqrt{5}}\right)$ with the positive direction of $x$-axis, upon reaching the ellipse surface, the ray is reflected from it. Slope of the reflected ray is

Let $5x-3y=8\sqrt{2}$ be normal at $P\left(\frac{5}{\sqrt{2}}, \frac{3}{\sqrt{2}}\right)$ to an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1, a>b$. If $M \& M^{\text{'}}$ are feet of perpendiculars from foci $S \& S^{\text{'}}$, respectively on the tangent at $P$, then the point of intersection of ${\mathrm{SM}}^{\text{'}}$ and ${\mathrm{S}}^{\text{'}}\mathrm{M}$ is

If the normals at $\alpha , \beta , \gamma$ and $\delta$ on an ellipse are concurrent, then the value of $\left(\sum _{}\cos \alpha \right)\left(\sum _{}\sec \alpha \right)$ is

The chords of contact of perpendicular tangents to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ touch another fixed ellipse whose equation is