Question

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 midpoint of the line segment joining the centers of $C_1$ and $C_2$ and $C$ be a circle touching circles $C_1$ and $C_2$ externally. If a common tangent to $C_{1}andC$ passing through P is also a common tangent to $C_{2}andC$, then find the radius of circle C.___

Answer 1

Since $\Delta$ADB is isosceles, if P is mid-point of AB, then $DP\perp AB$.


In $\Delta PFBcos\alpha =\frac{2\sqrt{2}}{3}\therefore sin\alpha =\frac{1}{3}In\Delta DEPtan\alpha =\frac{2\sqrt{2}}{R}orR=\frac{2\sqrt{2}}{tan\alpha }=8units$