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Let $C_{1} \equiv x^{2} + y^{2} - 169 = 0$ and $C_{2} \equiv x^{2} + y^{2} - 24 x - 32 y + 111 = 0.$ Let $A B$ be their common chord. Maximum area of $Δ A B C$ if $C$ lies on minor arc $A B$ (Of either of the circle), is equal to $a \sqrt{30} u n i t (a \in N) .$ Find the value of $a .$

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answer is 12.

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Let $M$ be the mid-point of $A B$
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$\Rightarrow A M^{2} = 17^{2} - x^{2} = 13^{2} - {(20 - x)}^{2}$
$\Rightarrow x = 13$
$\Rightarrow A B = 4 \sqrt{30},$ clearly when $C$ lies at R
Area is maximum $\Rightarrow$ Maximum area $= \frac{1}{2} \times 4 \sqrt{30} \times 6$
$= 12 \sqrt{30} s q . u n i t s$

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Let C1≡x2+y2−169=0 and C2≡x2+y2−24x−32y+111=0. Let AB be their common chord. Maximum area of ΔABC if C lies on minor arc AB (Of either of the circle), is equal to a30 unit(a∈N). Find the value of a.