Embibe Experts Solutions for Chapter: Ellipse, Exercise 1: EXERCISE-1

An ellipse is drawn with major and minor axes of lengths $10$ and $8$ respectively. Using one focus as centre, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. The radius of the circle is-

Which of the following is the common tangent to the ellipses $\frac{x^2}{a^2+b^2}+\frac{y^2}{b^2}=1$ and $\frac{x^2}{a^2}+\frac{y^2}{a^2+b^2}=1$?

Angle between the tangents drawn from point $(4, 5)$ to the ellipse $\frac{x^2}{16}+\frac{y^2}{25}=1$ is:

An ellipse is such that the length of the latus rectum is equal to the sum of the lengths of its semi principal axes. Then -

The eccentric angle of the point where the line, $5x-3y=8\sqrt{2}$ is a normal to the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ is -

$PQ$ is a double ordinate of the ellipse $x^2+9y^2=9$, the normal at $P$ meets the diameter through $Q$ at $R,$ then the locus of the mid point of $PR$ is -

If $F_1 \& F_2$ are the feet of the perpendiculars from the foci $S_1 \& S_2$ 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-

If $\tan {\theta }_1·\tan {\theta }_2=-\frac{a^2}{{ b}^2}$ then the chord joining two points ${\theta }_1 \& {\theta }_2$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{{ b}^2}=1$ will subtend a right angle at