Its distance from (0,0) is three times its distance from (4,0)?

Question

find an equation for the set of all points (x,y) satisfying the given conditions.

Answer 1

3√{(x - 4)² + y²} = √(x² + y²)
9{(x - 4)² + y²} = x² + y²
9(x² - 8x + 16) + 9y² = x² + y²
9x² - 72x + 144 + 9y² = x² + y²
8x² - 72x + 144 + 8y² = 0
x² - 9x + y² + 18 = 0
(x² - 9x + 81/4) + y² - 9/4 = 0
(x - 9/2)² + y² = 9/4

The locus of points is a circle with
center (9/2,0) and radius 3/2.