Find the radical centre of the three circles $x^{2} + y^{2} = a^{2}, (x - c)^{2} + y^{2} = a^{2}$ and $x^{2} + (y - b)^{2} = a^{2}$ .

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Solution

Here by using the formula
$S_{1} - S_{2} = 0, S_{2} - S_{3} = 0$ and $S_{3} - S_{1} = 0$ we will get the equation of radical axis and by solving them we can obtain required radical centre.
Radical axis of first and second circle is given by
$(x^{2} + y^{2}) - (x^{2} + y^{2} - 2 c x + c^{2}) = 0$
or $x = \frac{c}{2}$

Also, the radical axis of first and third circle is given by
$(x^{2} + y^{2}) - (x^{2} + y^{2} - 2 b y + b^{2}) = 0$
or $y = \frac{b}{2}$

Thus, the radical centre = $(\frac{c}{2}, \frac{b}{2})$ .

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