Consider two circles C 1 : x 2 + y 2 = 25 and C 2 : ( x - α ) 2 + y 2 = 16 , where α ∈ ( 5 , 9 ) . Let the…

Given,

$C_1:x^2+y^2=25, C_2:(x-\alpha {)}^2+y^2=16$ and $5<\alpha <9$

Now, plotting the diagram we get,

Also, given $\sin \theta =\frac{\sqrt{63}}{8}$

Now, from the above diagram, finding the area of  $\Delta \mathrm{OAP}$ we get, 

$\Delta \mathrm{OAP}=\frac{1}{2}\times \alpha \left(\frac{\beta }{2}\right)=\frac{1}{2}\times 5\times 4\sin \theta$

$\Rightarrow \alpha \beta =40\times \frac{\sqrt{63}}{8}$

$\Rightarrow \alpha \beta =5\times \sqrt{63}$

$\Rightarrow (\alpha \beta {)}^2=25\times 63=1575$