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Math Help - Equation of locus of two tangents

  1. #1
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    Equation of locus of two tangents

    If the normal at P(ap^2,2ap) to the parabola y^2=4ax meets the curve again at Q(aq^2,2aq) prove that p^2+pq+2=0. Prove that the equation of the locus of the point of intersection of he tangents to the parabola at P and Q is
    y^2(x+2a)+4a^3=0.

    I've done the first part of proving p^2+pq+2=0.
    for this second part, I tried to find a parametric equation for the locus the point of intersection of the tangents. Found the equation of the two tangents, then the point of intersection. The problem I have is there is no p or q in the equation I'm supposed to prove, while the point is in terms of p and q. Need someone to point me in the right direction and I should be able to finish off. Thanks!
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  2. #2
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    Quote Originally Posted by arze View Post
    If the normal at P(ap^2,2ap) to the parabola y^2=4ax meets the curve again at Q(aq^2,2aq) prove that p^2+pq+2=0. Prove that the equation of the locus of the point of intersection of he tangents to the parabola at P and Q is
    y^2(x+2a)+4a^3=0.

    I've done the first part of proving p^2+pq+2=0.
    for this second part, I tried to find a parametric equation for the locus the point of intersection of the tangents. Found the equation of the two tangents, then the point of intersection. The problem I have is there is no p or q in the equation I'm supposed to prove, while the point is in terms of p and q. Need someone to point me in the right direction and I should be able to finish off. Thanks!
    You should find that the point of intersection is P = (apq,a(p+q)). But you know that p^2+pq+2=0. Solve that for q, and substitute that value for q into the coordinates of P. That gives P = (-ap^2-2a,-2a/p). So if P = (x,y) then x = -ap^2-2a and y = -2a/p. Now eliminate p from those equations for x and y.
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  3. #3
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    Quote Originally Posted by Opalg View Post
    You should find that the point of intersection is P = (apq,a(p+q)). But you know that p^2+pq+2=0. Solve that for q, and substitute that value for q into the coordinates of P. That gives P = (-ap^2-2a,-2a/p). So if P = (x,y) then x = -ap^2-2a and y = -2a/p. Now eliminate p from those equations for x and y.
    Thanks! But how to eliminate p? Can you explain abit more? thanks
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    Quote Originally Posted by arze View Post
    Thanks! But how to eliminate p? Can you explain abit more? thanks
    Make p the subject in the parametric equation for y and substitute the resulting expression into the parametric equation for x.
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