Analytic Geometry

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Find the equation of a locus of a point which moves so that the sum of its distance from (2,0)and (-2,0) is 8.

it is about equation of a locus..
please help!!

  • Analytic Geometry -

    Your description defines an ellipse.
    does the question expect you to find that equation by using that definition?
    if so, then let such a point on that locus be P(x,y)

    √[(x-2)^2 + y^2] + √[(x+2)^2 + y^2] = 8
    √[(x-2)^2 + y^2] = 8 - √[(x+2)^2 + y^2]
    square both sides and expand
    x^2 - 4x + 4 + y^2 = 64 - 16√[(x+2)^2 + y^2] + x^2 + 4x + 4 + y^2
    16√[(x+2)^2 + y^2] = 64 + 8x
    2√[(x+2)^2 + y^2] = 8 + x
    squaring again and simplifying I get
    3x^2 + 4y^2 = 48
    divide each term by 48 to get it into standard form
    x^2/16 + y^2/12 = 1

    or, the easy way
    from the description 2a = 8 , a = 4
    (2,0) and (-2,0) must be the focal points so the midpoint or (0,0) must be the centre and c = 2
    since the focal points lie on the x-axis
    b^2 + c^2 = a^2
    b^2 + 4 = 16
    b^2 = 12

    standard form with centre (0,0) is
    x^2/a^2 + y^2/b^2 = 1
    so
    x^2/16 +y^2/12 = 1

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