Question

Show that the points $(3,2,2),(-1,1,3),(0,5,6)$ and $(2,1,2)$ lie on a sphere whose centre is $(1,3,4)$.

Answer 1

$A(3,2,2),B(-1,1,3),C(0,5,6),D(2,1,2)$

Centre$\left(O\right)=(1,3,4)$

Points on sphere are equidistant from sphere's centre.

$\therefore OA=\sqrt{{(3-1)}^2+{(2-3)}^2+{(2-4)}^2}=3\therefore OB=\sqrt{{(-1-1)}^2+{(1-3)}^2+{(3-4)}^2}=3\therefore OC=\sqrt{{(0-1)}^2+{(5-3)}^2+{(6-4)}^2}=3\therefore OD=\sqrt{{(2-1)}^2+{(1-3)}^2+{(2-4)}^2}=3$

$\therefore$ The points $A,B,C,D$ lie on a sphere with centre $O$.