Embibe Experts Solutions for Chapter: Ellipse, Exercise 1: JEE Main - 26 June 2022 Shift 2

A common tangent $T$ to the curves $C_1:\frac{x^2}{4}+\frac{y^2}{9}=1$ and $C_2:\frac{x^2}{42}-\frac{y^2}{143}=1$ does not pass through the fourth quadrant. If $T$ touches $C_1$ at $\left(x_1,y_1\right)$ and $C_2$ at $\left(x_2,y_2\right)$, then $\left|2x_1+x_2\right|$ is equal to _______.

Let the tangents at the points $P$ and $Q$ on the ellipse $\frac{x^2}{2}+\frac{y^2}{4}=1$ meet at the point $R\left(\sqrt{2},2\sqrt{2}-2\right)$. If $S$ is the focus of the ellipse on its negative major axis, then $S{P}^2+S{Q}^2$ is equal to

Let a line $L$ pass through the point of intersection of the lines $bx+10y-8=0$ and $2x-3y=0$, $b\in R-\left\{\frac{4}{3}\right\}$. If the line $L$ also passes through the point $\left(1,1\right)$ and touches the circle $17\left(x^2+y^2\right)=16$, then the eccentricity of the ellipse $\frac{x^2}{5}+\frac{y^2}{b^2}=1$ is

Let $S=\left\{\left(x,y\right)\in \mathrm{\mathbb{N}}\times \mathrm{\mathbb{N}}:9{\left(x-3\right)}^2+16{\left(y-4\right)}^2\le 144\right\}$
and $T=\left\{\left(x,y\right)\in \mathrm{ℝ}\times \mathrm{ℝ}:{\left(x-7\right)}^2+{\left(\mathrm{y}-4\right)}^2\le 36\right\}$
The $n\left(S\cap T\right)$ is equal to ______.

Let $C$ be the largest circle centred at $\left(2,0\right)$ and inscribed in the ellipse $\frac{x^2}{36}+\frac{y^2}{16}=1$. If $\left(1,\alpha \right)$ lies on $C$, then $10{\alpha }^2$ is equal to ______

Let a tangent to the curve $9x^2+16y^2=144$ intersect the coordinate axes at the points $A$ and $B$. Then, the minimum length of the line segment $AB$ is ______

If the maximum distance of normal to the ellipse $\frac{x^2}{4}+\frac{y^2}{b^2}=1,b<2$, from the origin is $1$ , then the eccentricity of the ellipse is:

The line $x=8$ is the directrix of the ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with the corresponding focus $\left(2,0\right)$. If the tangent to $E$ at the point $P$ in the first quadrant passes through the point $\left(0,4\sqrt{3}\right)$ and intersects the $x$-axis at $Q$, then ${\left(3PQ\right)}^2$ is equal to _____ .