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Math Help - Co-ordinate Geometry Parabola equation

  1. #1
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    Co-ordinate Geometry Parabola equation

    A line joining the points A(-4a,0) to the point P(a t^2,2at), where a is a positive constant.
    As t varies the locus of the midpoint of the line AP is a parabola C.

    a) Find the cartesian equation of C


    i dont quite know how to work this out and what deos it mean when the question says "As t varies"?
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  2. #2
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    Quote Originally Posted by Stylis10 View Post
    A line joining the points A(-4a,0) to the point P(a t^2,2at), where a is a positive constant.
    As t varies the locus of the midpoint of the line AP is a parabola C.

    a) Find the cartesian equation of C


    i dont quite know how to work this out and what deos it mean when the question says "As t varies"?
    1. Calculate the coordinates of the midpoint M\left(\dfrac{at^2-4a}2\ ,\ \dfrac{0+2at}2\right)

    2. You now have a curve in parametric form:

    c:\left\{\begin{array}{l}x=\dfrac{at^2-4a}2 \\ y = at\end{array}\right. ........ The variable for both equations is t (that means t varies!)

    3. Solve the 2nd equation for t and substitute t in the first equation by this term:

    x=\dfrac{a\left( \frac ya \right)^2-4a}2 ........ Rearrange this equation:

    x=\dfrac{y^2-4a^2}{2a}~\implies~\boxed{y^2=2ax+4a^2}

    Since a > 0 this is the equation of a parabola opening to the right.
    Last edited by earboth; October 19th 2008 at 07:56 AM.
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