The length of the common chord of the circles $(x - a)^{2} + (y - b)^{2} = c^{2}$ and $(x - b)^{2} + (y - a)^{2} = c^{2}$ is

\sqrt{4 c^{2} - 2 (a - b)^{2}}

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\sqrt{2 c^{2} - (a - b)^{2}}

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\sqrt{4 c^{2} - (a - b)^{2}}

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\sqrt{4 c^{2} + 2 (a - b)^{2}}

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Solution

The correct option is A $\sqrt{4 c^{2} - 2 (a - b)^{2}}$
$S_{1} : (x - a)^{2} + (y - b)^{2} = c^{2}$
$C_{1} = (a, b), r_{1} = c$
and $S_{2} : (x - b)^{2} + (y - a)^{2} = c^{2}$
$C_{2} = (b, a), r_{2} = c$
So, equation of common chord is $S_{1} - S_{2} = 0$
$- 2 a x + 2 b x - 2 b y + 2 a y = 0$
$(b - a) x - (b - a) y = 0$
Line $A B : x = y$ -----(1)
So, $C P = ∣ ∣ ∣ \frac{a - b}{\sqrt{2}} ∣ ∣ ∣$
So, In $Δ C P A$ , AP $= \sqrt{c^{2} - \frac{(a - b)^{2}}{2}}$
$= \sqrt{\frac{2 c^{2} - (a - b)^{2}}{2}}$
So, length of chord $A B = 2 A P$
$= \sqrt{4 c^{2} - 2 (a - b)^{2}}$

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