The locus of a point, which moves such that the sum of squares of...

Let $P\left(x, y\right)$ be the required point. Then,

${x}^{2} + {y}^{2} + ( x - 0{)}^{2} + ( y - 1{)}^{2} + ( x - 1{)}^{2} + ( y - 0{)}^{2} + ( x - 1{)}^{2} + ( y - 1{)}^{2} = 18$

$⇒ 4{x}^{2} + 4{y}^{2} - 4x - 4y + 4 = 18$

$⇒ {x}^{2} + {y}^{2} - x - y - \dfrac{14}{4} = 0$

Centre$≡ \left(\dfrac{1}{2}, \dfrac{1}{2}\right)$ and

Radius$, r = \sqrt{\dfrac{1}{4} + \dfrac{1}{4} + \dfrac{14}{4}} = 2units$

So,

$d = 2r = 2 \times 2 = 4$

$⇒ {d}^{2} = 16$