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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.

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asked by mandy
  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

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    posted by Reiny

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