Question

The centres of two circles $C_1$ and $C_2$ each of unit radius are at a distance of $6$ units from each other. Let $P$ be the mid point of the line segment joining the centres of $C_1$, and $C_2$ and $C$ be a circle touching $C_1$ and $C_2$ externally. If a common tangent to $C_1,$ and $C$ passing through $P$ is also a common tangent to $C_2$ and $C$, then the radius of the circle $C$ is:

Answer 1

Let $A_1,A_2$ and $M$ be the centres of the circles $C_1,C_2$ and $C$ respectively.
Let the common tangent through $P$ to $C_1$ and $C$ touch $C_1$ at $B_{1}andC$ at $B_2$ and it touches $C_2$ also at $B_2$
From right angled triangle $A_1{B}_2,P$
if $\underset{–}{A_{1}P{B}_2}=\alpha ,\frac{A_1{B}_2}{A_{1}P}=\frac{1}{3}$
$\Rightarrow \cos \alpha =\frac{2\sqrt{2}}{3}\Rightarrow P{B}_1=2\sqrt{2}=P{B}_2$
From triangle $MP{B}_2$
$\tan \alpha =\frac{P{B}_2}{M{B}_2}=\frac{2\sqrt{2}}{r}$
$\Rightarrow \frac{1}{2\sqrt{2}}=\frac{2\sqrt{2}}{r}$
$\Rightarrow r=8$
Ans: 8