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Math Help - Parametric Question

  1. #1
    Senior Member
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    Exclamation Parametric Question

    Hi

    P(2ap, ap^2), Q(2aq,aq^2) and R(2ar,ar^2) are points on the parabola x^2 = 4ay.
    a) Show that the equation of the normal at P is py+x = 2ap + ap^3.
    b) Find the coordinates of the point of intersection of the normals at P and Q.

    c) If the normals P,Q and R are constant, show that p +q +r=0.

    How on earth do you do this??

    Please help?
    Last edited by xwrathbringerx; January 22nd 2010 at 12:33 AM.
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  2. #2
    MHF Contributor
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    Quote Originally Posted by xwrathbringerx View Post
    Hi

    P(2ap, ap^2), Q(2aq,aq^2) and R(2ar,ar^2) are points on the parabola x^2 = 4ay.
    a) Show that the equation of the normal at P is py+x = 2ap + ap^3.
    b) Find the coordinates of the point of intersection of the normals at P and Q.

    c) If the normals P,Q and R are constant, show that p +q +r=0.

    How on earth do you do this??

    Please help?
    hi

    (a) Use the chain rule to get the gradient of tangent , then use m1m2=-1 for the gradient of normal

    x=2ap\Rightarrow \frac{dx}{dp}=2a

    y=ap^2\Rightarrow \frac{dy}{dp}=2ap

    so \frac{dy}{dx}=\frac{dy}{dp}\cdot \frac{dp}{dx}=2ap\cdot \frac{1}{2a}=p

    gradient of normal=-1/p

    Equation : y-ap^2=-\frac{1}{p}(x-2ap)

    Now simplify

    (b) Since Q is also a point on the parabola , its equation of normal would be qy+x=aq^3+2aq , just swap the 'p' with 'q' , then have them equal to solve for the point of intersection .
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