Question:
The locus of a point, which moves such that the sum of squares of its distances 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 ________.
Solution:
Let $\mathrm{P}(\mathrm{x}, \mathrm{y})$
$x^{2}+y^{2}+x^{2}+(y-1)^{2}+(x-1)^{2}+y^{2}+(x-1)^{2}+(y-1)^{2}$
$\Rightarrow 4\left(x^{2}+y^{2}\right)-4 y-4 x=14$
$\Rightarrow x^{2}+y^{2}-x-y-\frac{7}{2}=0$
$\mathrm{d}=2 \sqrt{\frac{1}{4}+\frac{1}{4}+\frac{7}{2}}$
$\Rightarrow \mathrm{d}^{2}=16$
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