Find the locus of midpoint of chord of $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ that passes through the origin.

x^{2} + y^{2} - g x - f y = 0

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x^{2} + y^{2} + g x + f y = 0

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x^{2} + y^{2} + 2 g x + 2 f y = 0

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x^{2} + y^{2} - 2 g x - 2 f y = 0

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Solution

The correct option is A $x^{2} + y^{2} + g x + f y = 0$
Let midpoint of the chord be $P (h, k)$ . So its equation is given by,
$T = S_{1} \Rightarrow h x + k y + g (x + h) + f (x + k) + c = h^{2} + k^{2} + 2 g h + 2 f k + c$
But it is given that it passes through $(0, 0)$
$\Rightarrow h^{2} + k^{2} + g h + k y = 0$
Hence locus of $P (h, k)$ is given by, $x^{2} + y^{2} + g x + f y = 0$

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