If the circles $x^{2} + y^{2} + 2 a x + 2 b y + c = 0$ and $x^{2} + y^{2} + 2 b x + 2 a y + c = 0$ touch each other then

(a - b)^{2} = 2 c

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a + b = 2 c

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(a + b)^{2} = 2 c

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(a + b)^{2} = c

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Solution

The correct option is C $(a + b)^{2} = 2 c$
$S_{1} : x^{2} + y^{2} + 2 a x + 2 b y + c = 0 \Rightarrow$ represents the circle with centre $C_{1} \equiv (- a, - b)$ & radius $r_{1} = \sqrt{a^{2} + b^{2} - c}$

$S_{2} : x^{2} + y^{2} + 2 b x + 2 a y + c = 0 \Rightarrow$ represents the circle with centre $C_{2} \equiv (- b, - a)$ & radius $r_{2} = \sqrt{a^{2} + b^{2} - c}$

Distance between centre $= \sqrt{(a - b)^{2} + (a - b)^{2}}$ $= \sqrt{2} (a - b)$
case - 1
$\sqrt{2} (a - b) = r_{1} + r_{2} = 2 \sqrt{a^{2} + b^{2} - c}$ ( touching externally )
$2 a^{2} + 2 b^{2} - 4 a b = 4 a^{2} + 4 b^{2} - 4 c$
$2 a^{2} + 2 b^{2} + 4 a b = 4 c$
$(a + b)^{2} = 2 c$

case - 2
$\sqrt{2} (a - b) = r_{1} - r_{2} = 0$ ( Touching internally ) which is not possible.

Hence, option C is correct.

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