The locus of a point, which moves such that the sum of squares of its distance from the points $(0, 0), (1, 0), (0, 1), (1, 1)$ is $18$ units, is a circle of diameter $d .$ Then $d^{2}$ is equal to

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Solution

Let the point be $P (h, k)$
$A (0, 0), B (1, 0), C (0, 1), D (1, 1)$
$(P A)^{2} + (P B)^{2} + (P C)^{2} + (P D)^{2} = 18$
$\Rightarrow h^{2} + k^{2} + (h - 1)^{2} + k^{2} + h^{2} + (k - 1)^{2} + (h - 1)^{2} + (k - 1)^{2} = 18$
$\Rightarrow 4 h^{2} + 4 k^{2} - 4 h - 4 k = 14$
$\Rightarrow h^{2} + k^{2} - h - k - \frac{7}{2} = 0$
$\Rightarrow x^{2} + y^{2} - x - y - \frac{7}{2} = 0$

$r = \sqrt{{(\frac{1}{2})}^{2} + {(\frac{1}{2})}^{2} + \frac{7}{2}}$
$\Rightarrow r = \sqrt{\frac{1 + 1 + 14}{4}} = 2$
$\Rightarrow d = 4$
$\Rightarrow d^{2} = 16$

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