Question

In the figure $PQ,P{O}_1$ and $O_{1}Q$ are the diameters of semicircles $C_1,C_2$ and $C_3$ with centres at $O_1,O_2$ and $O_3$ respectively and the circle $C_4$ touches the semicircles $C_1,C_2$ and $C_3$ . If $PQ=24$ units and the area of the circle $C_4$ is $A$ sq. units, then the value of $\frac{8\pi }{A}$ is equal to $\left(here,P{O}_1=O_{1}Q\right)$

Answer 1

Answer: 0.5

Let the point of contact of $C_4\&C_1$ is $A$ , center of $C_4$ is $O_4$ & radius is equal to $r$
$\Rightarrow A{O}_1=12\Rightarrow O_1{O}_4=12-r$
Also, $O_4{O}_3=r+6$ and $O_1{O}_3=6$
$\Rightarrow {\left(r+6\right)}^2={\left(12-r\right)}^2+36$
$\Rightarrow 36r=144\Rightarrow r=4\Rightarrow A=16\pi$
$\Rightarrow \frac{8\pi }{A}=\frac{8\pi }{16\pi }=\frac{1}{2}=0.5$