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Offline Centres

The centres of the circles are $(a, c)$ and $(b, c)$ and their radical axis is $y -$ axis. The radius of one of the circles is $r$ . The radius of the other circle is

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$r^{2} - a^{2} + b^{2}$

$2 (r^{2} - a^{2} + b^{2})$

$\sqrt{r^{2} - a^{2} + b^{2}}$

$2 \sqrt{r^{2} - a^{2} + b^{2}}$

answer is C.

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Detailed Solution

$C_{1} = (a, c), C_{2} = (b, c) r_{1} = r, r_{2} = ?$
O (0, 0) is a point on the radical axis
$\Rightarrow {oc}_{1}^{2} - r_{1}^{2} = {OC}_{2}^{2} - r_{2}^{2} \Rightarrow r_{2} = \sqrt{r^{2} - a^{2} + b^{2}}$

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