The equations of four circles are $(x \pm a)^{2} + (y \pm a)^{2} = a^{2}$ . The radius of a circle touching all the four circles externally is

2 \sqrt{2} a

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(\sqrt{2} + 1) a

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(\sqrt{2} - 1) a

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(2 + \sqrt{2}) a

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Solution

The correct option is B $(\sqrt{2} + 1) a$
From the given equation of circles

we find the centres at $(- a, a), (a, a), (a, - a), (- a, - a)$

${A C}^{2} = {2 a}^{2} + {2 a}^{2}$

$A C = 2 \sqrt{2} a$

now divide by $2$

the distance from centre of the small circle to the intersection of axes
will be $\sqrt{2 a}$

now add the radius of one small circle than the radius of circles which touch the small circle externally is

$\sqrt{2} a + a$

then optimal answer will be $a (1 + \sqrt{2})$ .

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