Three circles with centre A, B and C respectively, touch one another as shown in the figure. If A, B and C are collinear and PQ is a common tangent to the two smaller circles, where PQ = $4$ , the area of shaded region is

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Solution

Let radius of three circles be $R, r_{1}$ and $r_{2}$ of centre $A, B, C$ .
Let the end point of diameter be $Q R$ and tangent point $M$
$∠ Q P R = 90^{o}$ (angle in semicircle)
$P M = \frac{1}{2} M Q = 2$
In $△ P M R$
$(P M)^{2} + (M R)^{2} = (P R)^{2} 2^{2} + (2 r_{2})^{2} = (P R)^{2} 4 (1 + {r_{2}}^{2}) = (P R)^{2} (1)$
In $△ P M Q$
$(P M)^{2} + (M Q)^{2} = (P Q)^{2} 2^{2} + {r_{1}}^{2} = (P Q)^{2} (2)$
Adding $(1)$ and $(2)$
$8 + 4 (r_{1}^{2} + r_{2}^{2}) = (P R)^{2} + (P Q)^{2} 8 + 4 (r_{1}^{2} + r_{2}^{2}) = (Q R)^{2} 8 + 4 (r_{1}^{2} + r_{2}^{2}) = 4 R^{2}$
Area of shaded region $= π (R^{2} - (r_{1}^{2} + r_{2}^{2})) = 2 π$

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Three circles with centre A, B and C respectively, touch one another as shown in the figure. If A, B and C are collinear and PQ is a common tangent to the two smaller circles, where PQ =4, the area of shaded region is

Three circles with centre A, B and C respectively, touch one another as shown in the figure. If A, B and C are collinear and PQ is a common tangent to the two smaller circles, where PQ = $4$ , the area of shaded region is