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Tangent Circles

If the circle...

Question

If the circles $(x - a)^{2} + (y - b)^{2} = c^{2}$ , and $(x - b)^{2} + (y - a)^{2} = c^{2}$ touch each other , then $(a + b)^{2} =$

A

2 c^{2} + 4 a b

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B

4 c^{2}

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C

2 c^{2}

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D

2 c^{2} + 2 a b

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Solution

The correct option is C $2 c^{2} + 4 a b$
[ From point (1) ]
$S_{1} : (x - a)^{2} + (y - b)^{2} = c^{2}$
$C_{1} \equiv (a, b),$ radius $= c = r_{1}$
$S_{2} : (x - b)^{2} + (y - a)^{2} = c^{2}$
$c_{2} \equiv (b, a)$ radius $= c = r_{2}$
case - I
$S_{1}$ & $S_{2}$ touches each other externally then,
$c_{1} c_{2} = r_{1} r_{2}$
$\sqrt{2} (a - b) = r_{1} + r_{2} = 2 c .$
$(a - b) = \sqrt{2} c .$
$∴ (a + b)^{2} = (a - b)^{2} + 4 a b$
$(a + b)^{2} = 2 c^{2} + 4 a b$

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