Locus of mid-point of chord of the circle $x^{2} + y^{2} = r^{2}$ passing through (h, k).

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Solution

Let coordinates of Mid-point be $(l, m)$
Curve=> $x^{2} + y^{2} = r^{2}$
Equation of chord whose mid-points given=> $l x + m y - r^{2} = l^{2} + m^{2} - r^{2} => l x + m y = l^{2} + m^{2}$
$(h, k)$ lies on => $l h + m k = l^{2} + m^{2}$
for locus $l \to x, m \to y$
Required locus=> $h x + k y = x^{2} + y^{2} => x^{2} + y^{2} - h x - k y = 0$

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