Locus question that Im stuck on

Question

Show that the locus of a point, which moves so as always to be three times further from one fixed point than from another fixed point is a circle?

Answer 1

Hint: We assume we are working in the plane. Let the two given fixed points be $(a,b)$ and $(c,d)$.

By the Pythagorean Theorem, the distance of a point (x,y) from (a,b) is $$\sqrt{(x-a)^2+(y-b)^2}.$$ The distance of $(x,y)$ from $(c,d)$ is $$\sqrt{(x-c)^2+(y-d)^2}.$$ The distance condition says that $$\sqrt{(x-a)^2+(y-b)^2}=3\sqrt{(x-c)^2+(y-d)^2}.$$ Square both sides, bring all terms to one side, divide by $8$, You will get a familiar kind of equation.