Find equation of the circle, which passes through the origin, has its centre on the line $x + y = 4$ and cuts the circle $x^{2} + y^{2} - 4 x + 2 y + 4 = 0$ orthogonally

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Solution

Circle passes through origin
centre lies on $x + y = 4$
cuts the circle $x^{2} + y^{2} - 4 x + 2 y + 4 = 0$ orthogonally
Let circle be $x^{2} + y^{2} + 29 x + 2 f y + C = 0$
with centre $(- g, - f)$
$(0, 0)$ satisfies it
$C = 0$ _(1)
$- g - f = 4$ (2)
for orthogonality,
$2 g g^{'} + 2 f f^{'} = c + c^{'}$
$- 4 g + 2 f = 4$ __(3)
$g = - 2 f = - 2$ . [on solving (2), (3)]
circle : $x^{2} + y^{2} - 4 x - 4 y = 0$

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