Embibe Experts Solutions for Chapter: Ellipse, Exercise 1: JEE Advanced Paper 1 - 2020

Let $E$ be the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$. For any three distinct points $P,Q$ and $Q^{\text{'}}$ on $E$, let $M\left(P,Q\right)$ be the mid-point of the line segment joining $P$ and $Q$, and $M\left(P,Q^{\text{'}}\right)$ be the mid-point of the line segment joining $P$ and $Q^{\text{'}}$. Then the maximum possible value of the distance between $M\left(P,Q\right)$ and $M\left(P,Q^{\text{'}}\right)$, as $P,Q$ and $Q^{\text{'}}$ vary on $E$, is _____.

Let $a, b$ and $\lambda$ be positive real numbers. Suppose $P$ is an end point of the latus rectum of the parabola $y^2=4\lambda x$, and suppose the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ passes through the point $P$. If the tangents to the parabola and the ellipse at the point $P$ are perpendicular to each other, then the eccentricity of the ellipse is

Define the collections $\left\{E_1,\hspace{0.17em}{E}_2,\hspace{0.17em}{E}_3,....\right\}$ of ellipses and $\left\{R_1,\hspace{0.17em}{R}_2,\hspace{0.17em}{R}_3,....\right\}$ of rectangles as follows:

$E_1:\frac{x^2}{9}+\frac{y^2}{4}=1;$

$R_1:$ rectangle of largest area, with sides parallel to the axes, inscribed in $E_1;$

$E_n:$ ellipse $\frac{x^2}{a_n^2}+\frac{y^2}{b_n^2}=1$ of largest area inscribed in $R_{n-1},\hspace{0.17em}n>1;$

$R_{\mathrm{n}}:$ rectangle of largest area, with sides parallel to the axes, inscribed in $E_{\mathrm{n}},\mathrm{n}>1.$

Then which of the following options is/are correct?

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.