Find the image of the circle $x^{2} + y^{2} - 6 x - 8 y = 0$ in the line $3 x + 4 y + 25 = 0$ .

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Solution

Image of circle along line
$\Rightarrow$ $P$ line is Radical axis of circle.
$∴ S - S^{'} = 3 x + 4 y + 25$
$x^{2} + y^{2} - 6 x - 8 y - x^{2} - y^{2} - 2 q x - 2 f y - c = 3 x + 4 y + 25$
$(- 6 - 2 a) x + (- 8 - 2 S) y - (∴ 3 x + 4 y + 25$
$\Rightarrow \begin{matrix} 6 + g = - 3 g = - 9 / 2 \end{matrix} ∣ ∣ ∣ \begin{matrix} - 8 - 2 f = 4 f = - 6 \end{matrix} ∣ ∣ ∣ \begin{matrix} - c = 25 c = - 25 \end{matrix} ∣ ∣ ∣$
$∴ x^{2} + y^{2} - 9 x - 12 y - 21 = 0$

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