Amit M Agarwal Solutions for Chapter: Ellipse, Exercise 2: Work Book Exercise 15.2

A tangent to the ellipse $4x^2+9y^2=36$ is cut by the tangent at the extremities of the major axis at $T$ and $T^{\text{'}}$. The circle on $T{T}^{\text{'}}$ as diameter passes through the point

If $PSQ$ is a focal chord of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ $a>b,$ then the harmonic mean of $SP$ and $SQ$ is

Tangent is drawn to the ellipse $\frac{x^2}{27}+y^2=1$ at $\left(3\sqrt{3}\cos \theta ,\sin \theta \right),$ where $\theta \in \left(0,\frac{\pi }{2}\right),$ then the value of $\theta$ such that sum of intercepts on axes made by the tangent is minimum, is

If $F_1$ and $F_2$ are the foot 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 $\left(S_1{F}_1\right)\left(S_2{F}_2\right)$ is equal to

The slope of a common tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and a concentric circle of radius $r$ is

If $p$ and $p^{\text{'}}$ denote the lengths of the perpendicular from a focus and the centre of an ellipse with semi-major axis of length $a$, respectively on a tangent to the ellipse and $r$ denotes the focal distance of the point, then

Tangents are drawn to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ $(a>b)$ and the circle $x^2+y^2=a^2$ at the points, where a common ordinate cuts them on the same side of the $X$-axis. Then, the greatest acute angle between these tangents is given by