The equation to the locus of the midpoints of chords of the circle $x^{2} + y^{2} = r^{2}$ having a constant length 2l is

x^{2} + y^{2} = l^{2}

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x^{2} + y^{2} = r^{2} - l^{2}

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x^{2} + y^{2} = r^{2} + l^{2}

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x^{2} + y^{2} = 4 l^{2}

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Solution

The correct option is B $x^{2} + y^{2} = r^{2} - l^{2}$

Let $P (x_{1}, y_{1})$ be a point on the locus
Then P is the mid point of a chord AB of the circle $x^{2} + y^{2} = r^{2}$
Now $O P^{2} + A P^{2} = O A^{2} \Rightarrow x_{1}^{2} + y_{1}^{2} = r^{2} - l^{2}$
$∴ T h e l o c u s o f P i s x^{2} + y^{2} = r^{2} - l^{2}$

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