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

A series of concentric ellipses $E_1,E_2,\dots \dots ,E_n$ are drawn such that $E_n$ touches the extremities of the major axis of $E_{n-1}$ and the foci of ${\dot{E}}_n$ coincide with the extremities of minor axis of $E_{n-1} .$ If the eccentricity of the ellipses is independent of $n,$ then the value of the eccentricity, is

If the normal at any point $P$ on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the auxiliary circle at $Q$ and $R$, such that $\angle QOR=90°$, where $O$ is the center of the ellipse, then

$P \text{\&} Q$ are corresponding points on the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$, and the auxiliary circle respectively. The normal at $P$ to the ellipse meets $CQ$ in $R$ where $C$ is centre of the ellipse. Then $\ell \left(CR\right)$ is

Identify correct statement(s) about conic $\sqrt{{\left(x-5\right)}^2+{\left(y-7\right)}^2}+\sqrt{{\left(x+1\right)}^2+{\left(y+1\right)}^2}=12$

In a triangle$ABC$ with fixed base $BC,$ the vertex $A$ moves such that $\cos B+\cos C=4{\sin }^2\frac{A}{2}.$ If $a, b$ and $c$ denote the lengths of the sides of the triangle opposite to the angles $A, B$ and $C$, respectively, then

Let $P$  be a variable point on the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ with foci at $F_{1 }$ & $F_2$

$P$ and $Q$ are two points on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ $\left(a, b>0\right)$ whose eccentric angles differ by ${90}^{°}$, then

$AB$ and $CD$ are two equal and parallel chords of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. Tangents to the ellipse at $A$ and $B$ intersect at $P$ and tangents at $C$ and $D$ at $Q$. The line $PQ$