Find the radical centre of the sets of circles
$x^{2} + y^{2} + x + 2 y + 3 = 0, x^{2} + y^{2} + 2 x + 4 y + 5 = 0,$ and $x^{2} + y^{2} - 7 x - 8 y - 9 = 0.$

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Solution

The equation of the radical axis for a pair of circles $S_{1}$ and $S_{2}$ is given by
$S_{1} - S_{2} = 0$ . In the above case $S_{1} = 0 \to x^{2} + y^{2} - x y + 6 x - 7 y + 8 = 0$ and $S_{2} = 0 \to x^{2} + y^{2} - x y - 4 = 0$ .
Hence $S_{1} - S_{2} = 0$ implies $6 x - 7 y + 12 = 0$
Hence the equation of the radical axis is $6 x - 7 y + 12 = 0$

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