Embibe Experts Solutions for Chapter: Ellipse, Exercise 1: TS EAMCET - 5 August 2021 Shift 1

A focus of an ellipse having eccentricity $\frac{1}{2}$ is at $\left(0, 0\right)$ and a directrix is the line $x=4.$ Then the equation of one such ellipse is

The area (in square units) of the quadrilateral formed by joining the focii of the two ellipses $\frac{x^2}{9}+\frac{y^2}{5}=1$ and $\frac{x^2}{5}+\frac{y^2}{9}=1$ is

$P$ is a point on the conic $a^2{x}^2+b^2{y}^2=a^2\left(a^2+b^2-y^2\right)$ and $S$ is a focus of that conic. $M$ is the foot of the perpendicular from $P$ on to a directrix of that conic nearer to $S$. If $PM=KSP$, then $K=$

For the ellipse $4{\left(x-2y+1\right)}^2+9{\left(2x+y+2\right)}^2=25,$ which of the following is true?

An ellipse has its major axis along $y$-axis and minor axis along $x$-axis. If its length of latus rectum is $\frac{2}{3}$ times of its minor axis, then the eccentricity of the ellipse is

If the length and breadth of a rectangle of maximum area that can be inscribed in an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ are $8\sqrt{2}$ and $4\sqrt{2}$ respectively, then the eccentricity of that ellipse is

A variable tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ makes intercepts on both the axes. The locus of the middle point of the portion of the tangent between the coordinate axes is

If a circle $\left(x-1\right){}^2+y^2=r^2$ touches the ellipse $x^2+4y^2=16$ internally, then $r=$