If a circle passes through the point $(a, b)$ and cuts the circle $x^{2} + y^{2} = 4$ orthogonally then the locus of its centre is -

2 a x + 2 b y + a^{2} + b^{2} + 4 = 0

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2 a x + 2 b y - (a^{2} + b^{2} + 4) = 0

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2 a x - 2 b y + a^{2} + b^{2} + 4 = 0

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2 a x - 2 b y - (a^{2} + b^{2} + 4) = 0

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Solution

The correct option is B $2 a x + 2 b y - (a^{2} + b^{2} + 4) = 0$
Let the variable circle be $x^{2} + y^{2} + 2 y x + 2 f y + c = 0$ ...(1)
Circle (1) cuts the circle $x^{2} + y^{2} - 4 = 0$ orthoginally
$∴ 2 g .0 + 2 f .0 = c - 4 \Rightarrow c = 4$
Since circle (1) passes through $(a, b)$ .
$∴ a^{2} + b^{2} + 2 g a + 2 f b + c = 0$
Therefore locus of center $(- g, - f)$ is
$2 a x + 2 b y - (a^{2} + b^{2} + 4) = 0$

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