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Math Help - Analytic Geometry

  1. #1
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    Question Analytic Geometry

    Hello, everybody. I need help...
    Show that, for all values of p, the point P given by x=ap2, y=2ap lies on the curve y2=4ax.
    a) Find the equation of this normal to this curve at the point P.
    If this normal meets the curve again at the point Q (aq2, 2aq). Show that p2+pq+2=0
    b) Determine the coordinates of R, the point of intersection of the tangents of the curve at the point P and Q.
    Hence, show that the locus of the point R is y2(x+2a) +4a3=0

    I already solved Q (a) and first question of Q (b). However, I can’t solve the last question: “Hence, show that the locus of the point R is y2(x+2a) +4a3=0” in Q (b). Can somebody help me?
    Thanks!!
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Analytic Geometry

    Presumably, you have found that that R has the coordinates:

    (apq,a(q+p))

    Now, use the relationship between p and q you found earlier:

    p^2+pq+2=0

    which when solved for q is:

    q=-\frac{p^2+2}{p}

    Now you may write the coordinates of R as parametric equations in one parameter, which you may then eliminate to obtain the required Cartesian equation.
    Last edited by MarkFL; September 28th 2012 at 08:35 PM.
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  3. #3
    MHF Contributor
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    Re: Analytic Geometry

    Hey AuXian.

    I haven't seen this stuff since high school (more than 10 years) but the wiki page gives a derivation of the locus for the parabola:

    Parabola - Wikipedia, the free encyclopedia

    I'm not sure if you just formulas or have to derive things, but if you just formulas the wiki gives a derivation from start to finish in terms of the a term.
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