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Math Help - locus of parabola problems

  1. #1
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    locus of parabola problems

    a) The normals to the parabola x^2 = 4ay at the points P(2ap, ap^2) and Q(2aq, aq^2) are perpendicular to each other and meet at R. Prove that R lies on the parabola x^2 = a(y-3a)

    b) P is a variable point on the parabola x^2=4ay and N is the foot of the perpendicular drawn from the focus S to the normal at P. Show that the locus of N is x^2 = a(y-a)

    if anyone could help me with these problems, i would REALLY appreciate it, thankYOU !
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  2. #2
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    Quote Originally Posted by flyinhigh123 View Post
    a) The normals to the parabola x^2 = 4ay at the points P(2ap, ap^2) and Q(2aq, aq^2) are perpendicular to each other and meet at R. Prove that R lies on the parabola x^2 = a(y-3a)

    b) P is a variable point on the parabola x^2=4ay and N is the foot of the perpendicular drawn from the focus S to the normal at P. Show that the locus of N is x^2 = a(y-a)

    if anyone could help me with these problems, i would REALLY appreciate it, thankYOU !
    The gradient of the normals at P and Q are -p and -q, so the normals are perpendicular gives pq = -1.

    Now use the technique I showed you in my posting to your previous question (where pq = -4). Use the same equations for x and y - the ones that represent the point of intersection of the normals; eliminate pq in the equation for x; square both sides; then form an expression for y in terms of (p^2+q^2) and pq; then eliminate p and q altogether, and you're done.

    Grandad
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