First slide

Analysis of two circles in circles

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

Moderate

A

$a = b \pm 2 c$
B

$a = b \pm \sqrt{2} c$
C

$a = b \pm c$
D

none of these

Solution

The circles will touch each other if the length of the common chord is zero i.e.
$\sqrt{4 c^{2} - 2 (a - b)^{2}} = 0 \Rightarrow 2 c^{2} = (a - b)^{2} \Rightarrow a - b = \pm \sqrt{2} c$

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