Embibe Experts Solutions for Chapter: Ellipse, Exercise 1: Level 1

If the eccentricity of the ellipse $\frac{x^2}{a^2+1}+\frac{y^2}{a^2+2}=1$ is $\frac{1}{\sqrt{6}},$ then the latus rectum of the ellipse is

In an ellipse, the distance between the foci is $8$ and the distance between the directrices is $25$. The length of the major axis is

What is the sum of the major and minor axes of the ellipse, whose eccentricity is $\frac{4}{5}$ and length of latus rectum is $14.4$ units?

The minimum area of the triangle formed by any tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with the coordinate axes is

Let $L$ be a common tangent line to the curves $4x^2+9y^2=36$ and ${\left(2x\right)}^2+{\left(2y\right)}^2=31$. Then the square of the slope of the line $L$ is ______.

If $\left(\sqrt{3}\right)bx+ay=2ab$ touches the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ then the eccentric angle of the point of contact is

The equation of the tangents to the ellipse $9x^2+16y^2=144$ which passes through the point $\left(2,3\right)$ 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