Embibe Experts Solutions for Chapter: Ellipse, Exercise 2: EXERCISE-2

If the chord through the points whose eccentric angles are $\theta \& \varphi$ on the ellipse, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$passes through the focus, then the value of $\tan \left(\frac{\theta }{2}\right)\tan \left(\frac{\varphi }{2}\right)$ is -

If point $P(\alpha +1,\alpha )$ lies between the ellipse $16x^2+9y^2-16x=0$ and its auxiliary circle, then -

where [.] denotes greatest integer function.

If latus rectum of an ellipse $\frac{x^2}{16}+\frac{y^2}{b^2}=1 \{0<b<4\}$, subtend angle $2\theta$ at farthest vertex such that $cosec\theta =\sqrt{5}$, then -

If $x-2y+k=0$ is a common tangent to $y^2=4x$ & $\frac{x^2}{a^2}+\frac{y^2}{3}=1\left(a>\sqrt{3}\right)$, then the value of $a, k$ and other common tangent are given by -

All ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 \left(0<b<a\right)$ has fixed major axis. Tangent at any end point of latus rectum meet at a fixed point which can be -

Eccentric angle of a point on the ellipse $x^2+3y^2=6$ at a distance $\sqrt{3}$ units from the centre of the ellipse is -

For the ellipse $9x^2+16y^2-18x+32y-119=0$, which of the following is/are true -

With respect to the ellipse $4x^2+7y^2=8$, the correct statement(s) is/are -