If from the origin a chord is drawn to the circle $x^{2} + y^{2} - 2 x = 0,$ then the locus of the mid point of the chord has equation

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$x^{2} + y^{2} + x + y = 0$

$x^{2} + y^{2} - 2 x + y = 0$

$x^{2} + y^{2} - x = 0$

$x^{2} + y^{2} + 2 x + y = 0$

answer is C.

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Detailed Solution

Let $(h, k)$ be the mid point of the chord drawn through the origin. Then the equation of the chord is

$h x + k y - (x + h) = h^{2} + k^{2} - 2 h$ [Using $: T = S'$ ]

This passes through $(0, 0) .$

$∴ - h = h^{2} + k^{2} - 2 h \Rightarrow h^{2} + k^{2} - h = 0$

Hence, the locus of $(h, k)$ is $x^{2} + y^{2} - x = 0$

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