The locus of a point, which moves such that the sum of squares of its distances from the p…

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 \_\_\_\_\_\_\_\_.

Answer: 16

Accepted Answer

$$x^2+y^2+[x^2+(y-1)^2]+[(x-1)^2+y^2]+[(x-1)^2+(y-1)^2]=18$$
$$\Rightarrow4(x^2+y^2)-4x-4y=14\ \Longrightarrow\ x^2+y^2-x-y-\frac72=0.$$

$$r=\sqrt{\frac14+\frac14+\frac72}=2,\qquad d=2r=4,\qquad d^2=16.$$

Solutions · 1