From the origin chords are drawn to the circle $(x - 1)^{2} + y^{2} = 1$ . The equation of the locus of the middle points of these chords is

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Solution

The correct option is C

The given circle is $x^{2} + y^{2} - 2 x = 0$ . Let $(x_{1}, y_{1})$ be the middle point of any chord of this circle, than its equation is $S_{1} = T$ .

or $x_{1}^{2} + y_{1}^{2} - 2 x_{1} = x x_{1} + y y_{1} - (x + x_{1})$

If it passes through (0,0), then

$x_{1}^{2} + y_{1}^{2} - 2 x_{1} = - x_{1} \Rightarrow x_{1}^{2} + y_{1}^{2} - x_{1} = 0$

Hence the required locus of the given point $(x_{1}, y_{1})$ is $x^{2} + y^{2} - x = 0$

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