Consider two circles and , where . Let the angle between the two radii (one to each circle) drawn from one of the intersection points of and be . If the length of common chord of and is , then the value of equals __________. [2024]

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Solution

Q.
Consider two circles $C_{1} : x^{2} + y^{2} = 25$ and $C_{2} : {(x - α)}^{2} + y^{2} = 16$ , where $α \in (5, 9)$ . Let the angle between the two radii (one to each circle) drawn from one of the intersection points of $C_{1}$ and $C_{2}$ be $\sin^{- 1} (\frac{\sqrt{63}}{8})$ . If the length of common chord of $C_{1}$ and $C_{2}$ is $β$ , then the value of ${(α β)}^{2}$ equals __________. [2024]

Ans.
(1575)

We have,

$C_{1} : x^{2} + y^{2} = 25$ or $C_{2} : {(x - α)}^{2} + y^{2} = 16$

Let $θ$ be the angle between two radii

$∴ θ = \sin^{- 1} \frac{\sqrt{63}}{8}$

$\Rightarrow \sin θ = \frac{\sqrt{63}}{8}$

Area of $△ O A B = \frac{1}{2} \times 4 \times 5 \sin θ$

$\Rightarrow \frac{1}{2} \times α \times \frac{β}{2} = 10 \frac{\sqrt{63}}{8} \Rightarrow α β = 5 \sqrt{63} \Rightarrow {(α β)}^{2} = 1575$ .

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