Question

In the figure $PQ,PO_{1}$ 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

Let the point of contact of $C_{4} \, \& \, C_{1}$ is $A$ , center of $C_{4}$ is $O_{4}$ & radius is equal to $r$
$\Rightarrow AO_{1}=12\Rightarrow O_{1}O_{4}=12-r$
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 $
$\Rightarrow \frac{8 \pi }{A}=\frac{8 \pi }{16 \pi }=\frac{1}{2}=0.5$