A point moves such that its distance from point A(1,0) is twice its distance from B(-2,0).Find the equation of the locus of the point.

Answer 1

let the point be P(x,y)
then PA = 2PB
√( (x-1)^2 + y^2) = 2√( (x+2)^2 + y^2)
square both sides and expand
x^2 - 2x + 1 + y^2 = 4(x^2 + 4x + 4 + y^2)
x^2 - 2x + 1 + y^2 = 4x^2 + 16x + 16 + 4y^2
3x^2 + 18x + 3y^2 = -15
x^2 + 6x + y^2 = -5
complete the square:
x^2 + 6x + 9 + y^2 = -5+9
(x+3)^2 + y^2 = 4
circle with centre (-3,0) and radius 2

confirmed here:
http://www.wolframalpha.com/input/?i=plot+√%28+%28x-1%29%5E2+%2B+y%5E2%29+%3D+2√%28+%28x%2B2%29%5E2+%2B+y%5E2%29