There are two circles $$C_{1}$$ and $$C_{2}$$ of radii 3 and 8 units respectively. The common internal tangent, T, touches the circles at points $$P_{1}$$ and $$P_{2}$$ respectively. The line joining the centers of the circles intersects T at X. The distance of X from the center of the smaller circle is 5 units. What is the length of the line segment $$P_{1} P_{2}$$ ?

Given : $$OP_1 = 3 , O'P_2 = 8 , OX = 5$$ units

To find : $$P_1P_2 = ?$$

Solution : In $$\triangle OP_1X$$

=> $$(P_1X)^2 = (OP_1)^2 - (OX)^2$$

=> $$(P_1X)^2 = 5^2 - 3^2 = 25 - 9$$

=> $$P_1X = \sqrt{16} = 4$$

In $$\triangle OP_1X$$ and $$\triangle O'P_2X$$

$$\angle OXP_1 = O'XP_2$$   (Vertically opposite angles)

$$\angle OP_1X = O'P_2X = 90$$

=> $$\triangle OP_1X \sim \triangle O'P_2X$$

=> $$\frac{XP_1}{XP_2} = \frac{OP_1}{O'P_2}$$

=> $$XP_2 = 4 \times \frac{8}{3} = 10.66$$

$$\therefore P_1P_2 = P_1X + XP_2 = 4 + 10.66 = 14.66$$ units