Question

Let $T$ be the line passing through the points $P(-2,7)$ and $Q(2,-5)$. Let $F_1$ be the set of all pairs of circles $(S_1,S_2)$ such that $T$ is tangent 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 $(S_1,S_2)$ 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 $R(1,1)$ 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?

• A. The point $(-2,7)$ lies in $E_1$.
• B. The point $(\frac{4}{5},\frac{7}{5})$ does $\text{NOT}$ lie in $E_2$.
• C. The point $(\frac{1}{2},1)$ lies in $E_2$.
• D. The point $(0,\frac{3}{2})$ does $\text{NOT}$ lie in $E_1$.

Answer 1

Answer: (B), (D)

The correct options are
B The point $(\frac{4}{5},\frac{7}{5})$ does $\text{NOT}$ lie in $E_2$.
D The point $(0,\frac{3}{2})$ does $\text{NOT}$ lie in $E_1$.


$PL=ML[\because △O_{1}PL\cong △O_{1}ML]$
$\text{and}ML=QL[\because △O_{2}ML\cong △O_{2}QL]$
$\Rightarrow PL=QL=ML$
$\Rightarrow L$ is the mid-point of $PQ.$
Thus, coordinates of $L$ is $(0,1).$
Let coordinates of $M$ be $(x,y).$
$\because ML=PL\Rightarrow (x-0{)}^2+(y-1{)}^2=40$
Therefore, locus of $M$ is a circle with centre $(0,1)$ and radius $\sqrt{40}.$
$\therefore E_1=\left\{\right(x,y):(x-0{)}^2+(y-1{)}^2=40\}$
Clearly, $(-2,7)$ does not lie in $E_1.$
Also, $(0,\frac{3}{2})$ does not lie in $E_1.$


Locus of points in $E_2$ is a circle with coordinates of diameter $L(0,1)$ and $R(1,1).$
$\therefore E_2=\left\{\right(x,y):(x-0\left)\right(x-1)+(y-1\left)\right(y-1)=0\}$
$\text{i.e,}{E}_2=\left\{\right(x,y):x(x-1)+(y-1{)}^2=0\}$
$\Rightarrow (\frac{4}{5},\frac{7}{5}),(\frac{1}{2},1)$ does not lie in $E_2.$