Embibe Experts Solutions for Chapter: Circle, Exercise 1: JEE Advanced Paper 1 - 2021

Let $T$ be the line passing through the points $P\left(-2,7\right)$ and $Q\left(2,-5\right)$. Let $F_1$ be the set of all pairs of circles $\left(S_1,S_2\right)$ such that $T$ is tangents to $S_1$ at $P$ and tangent to $S_2$ at $Q$, and also such that $S_1$ and $S_2$ touch each other at a point, say, $M$. Let $E_1$ be the set representing the locus of $M$ as the pair $\left(S_1, S_2\right)$ varies in $F_1$. Let the set of all straight line segments joining a pair of distinct points of $E_1$ and passing through the point $R\left(\mathrm{1,1}\right)$ be $F_2$ . Let $E_2$ be the set of the mid-points of the line segments in the set $F_2$ . Then, which of the following statement(s) is (are) TRUE?

Let $E_1$ and $E_2$ be two ellipse whose centers are at the origin. The major axes of $E_1 \& E_2$ lie along the $x$-axis and the $y$-axis, respectively. Let $S$ be the circle $x^2+{\left(y-1\right)}^2=2.$ The straight line $x+y=3$ touches the curves $S, E_1$ and $E_2$ at $P,Q$ and $R,$ respectively. Suppose that $PQ=PR= \frac{2\sqrt{2}}{3}$. If$e_1$ and $e_2$ are the eccentricities of $E_1$ and $E_2,$ respectively, then the correct expression(s) is(are)

Consider a triangle $\mathrm{\Delta }$ whose two sides lie on the $x$-axis and the line $x+y+1=0$. If the orthocenter of $\mathrm{\Delta }$ is $(1,1)$, then the equation of the circle passing through the vertices of the triangle $\mathrm{\Delta }$ is

A line $y=mx+1$ intersects the circle ${\left(x-3\right)}^2+{\left(y+2\right)}^2=25$ at the points $P$ and $Q.$ If the midpoint of the line segment $PQ$ has $x-$coordinate $-\frac{3}{5},$ then which one of the following options is correct?

Let the point $B$ be the reflection of the point $A\left(2, 3\right)$ with respect to the line $8x-6y-23=0.$ Let $T_A$ and $T_B$ be circle of radii $2$ and $1$ with centres $A$ and $B$ respectively. Let $T$ be a common tangent to the circle $T_A$ and $T_B$ such that both the circle are on the same side of $T.$ If $C$ is the point of intersection of $T$ and the line passing through $A$ and $B,$ then the length of the line segment $AC$ is ___________.

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

Let $RS$ be the diameter of the circle $x^2+y^2=1$ where, $S$ is the point $\left(1,0\right)$. Let $P$be a variable point (other than $R \& S$) on the circle and tangents to the circle at $S \& P$meet at the point $Q$. The normal to the circle at $P$ intersects a line drawn through $Q$ parallel to $RS$ at point $E$ . Then, the locus of $E$ passes through the point (s):

The circle $C_1:x^2+y^2=3,$ with centre at O, intersects the parabola $x^2=2y$ at the point P in the first quadrant. Let the tangent to the circle $C_1$ at P touches other two circles $C_2$ and $C_3$ at $R_2$ and $R_3$ , respectively. Suppose $C_2$ and $C_3$ have equal radii $2\sqrt{3}$ and centres $Q_2$ and ${ Q}_3$ , respectively. If $Q_2$ and $Q_3$ lie on the y - axis, then