Question

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

Answer 1

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

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