A variable chord is drawn through the origin to the circle $x^{2} + y^{2} - 2 a x = 0$ . The locus of the centre of the circle drawn on the chord as diameter is?

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

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

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

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

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Solution

The correct option is C $x^{2} + y^{2} - a x = 0$
Given
$x^{2} + y^{2} - 2 a x = 0 ⟹ x^{2} + y^{2} + a^{2} - 2 a x - a^{2} = 0 ⟹ (x - a)^{2} + y^{2} = a^{2}$

So the center of circle is $O (a, 0)$ and radius $= a$

Let the center of chord drawn through be $P (h, k)$

Let the other end of chord be $(c, b)$

So, $(\frac{c + 0}{2}, \frac{b + 0}{2})$ = $(h, k)$

$⟹ c = 2 h, b = 2 k$

But point $(c, b)$ lies on the circle

$⟹ x^{2} + y^{2} - 2 a x = 0$

So, $(2 h)^{2} + (2 k)^{2} - 2 a .2 h = 0$

$⟹ h^{2} + k^{2} - a h = 0$

Hence Its locus is $x^{2} + y^{2} - a x = 0$

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