Consider two circles C _ 1 : x ^ 2 + y ^ 2 = 25 and C _ 2 : ( x - α...

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 $sin\theta = \dfrac{\sqrt{63}}{8}$

Now, from the above diagram, finding the area of $\delta OAP$ we get, 

$\delta OAP = \dfrac{1}{2} \times \alpha\left(\dfrac{\beta}{2}\right) = \dfrac{1}{2} \times 5 \times 4sin\theta$

$⇒ \alpha\beta = 40 \times \dfrac{\sqrt{63}}{8}$

$⇒ \alpha\beta = 5 \times \sqrt{63}$

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