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Math Help - Conditions for Parametric Parabolas

  1. #1
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    Exclamation Conditions for Parametric Parabolas

    Hi guys

    P and Q are the end points of a focal chord of the parabola x^2 = 4ay with focus S. If the coordinates of P and Q are(2ap, ap^2) and (2aq, aq^2) respectively.
    (a) Write down the gradients of PS and QS. \frac{p^2-1}{2p} and \frac{q^2-1}{2q} respectively
    (b) Show that pq = -1
    (c) Find the coordinates of the midpoint R of PQ in terms of p. [a(p-\frac{1}{p}), \frac{a}{2}(P^2+\frac{1}{p^2}]
    (d) Show that the equation of the locus of R is  x^2 = 2a(y-a).

    I've done what the question has asked me to do BUT my teacher wants me to go one step further and find the condition for the locus of R. Could someone please explain this to me - I dont geddit from the notes I've been given

    Thanx a lot
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  2. #2
    MHF Contributor

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    Parabolas have the property that the distance from the focus to any point on the parabola is the same as the distance from that point to the directrix.

    For a parabola through (0,0) with vertical axis, opening upward, if the focus is (0, c), then the directrix is the horizontal line y= -c. You need to find the focus and directrix of this parabola in order to give the "condition" for the parabola.
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  3. #3
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    Exclamation

    wat exactly do i do when i find the directrix + focus???
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