The centers 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 centers 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 $C_{1}$ and $C_{2}$ then the radius of the circle C is

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answer is 8.

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Detailed Solution

Let $A_{1}, A_{2}$ and M be the centers 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 $B_{1}$ , C at B₂ and $C_{2}$ also at $B_{2}$ .
Question Image
From right angled triangle $A_{1} B_{1} P$
If $A_{1} P B_{1} = α, \sin α = \frac{A_{1} B_{1}}{A_{1} P} = \frac{1}{3}$
$\Rightarrow \cos α = \frac{2 \sqrt{2}}{3} \Rightarrow P B_{1} = 2 \sqrt{2} = P B_{2}$
From triangle $M P B_{2}$
$\tan α = \frac{P B_{2}}{M B_{2}} = \frac{2 \sqrt{2}}{r} \Rightarrow \frac{1}{2 \sqrt{2}} = \frac{2 \sqrt{2}}{r} \Rightarrow r = 8$