The locus of the middle points of those chords of the circle $x^{2} + y^{2} = 4$ which subtend a right angle at the origin is

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Solution

The correct option is C

Let the mid-point of chord is (h,k). Also radius of circle is 2. Therefore

$\frac{O C}{O B} = c o s 45^{o} \Rightarrow \frac{\sqrt{h^{2} + k^{2}}}{2} = \frac{1}{\sqrt{2}} \Rightarrow h^{2} + k^{2} = 2$

Hence locus is $x^{2} + y^{2} = 2$

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From origin, chords are drawn to the circle $x^{2} + y^{2} - 2 y = 0$ .The locus of the middle points of these chords is

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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