A point B moves on the curve $x^{2} + y^{2} + x + y = 0$ and A is $(1, 2)$ . Find the locus of the midpoint of line joining A and B.

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Solution

$A (1, 2)$ and let B be $(x, y)$ lies on $x^{2} + y^{2} + x + y = 0$
Let the mid-point of AB be $(h, k) = (\frac{x + 1}{2}, \frac{2 + y}{2}) => (x, y) = ((2 h - 1), (2 k - 2))$
$(x, y)$ lies on $x^{2} + y^{2} + x + y = 0$
$=> {(2 h - 1)}^{2} + {(2 k - 2)}^{2} + (2 h - 1) + (2 k - 2) = 0 => 4 h^{2} + 4 k^{2} - 2 h - 6 k + 4 - 2 = 0 => 2 h^{2} + 2 k^{2} - h - 3 k + 1 = 0 l o c u s => 2 x^{2} + 2 y^{2} - x - 3 y + 1 = 0$

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