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Math Help - Parabola question

  1. #1
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    Post Parabola question

    So, okay I don't even know what the question means. I'm so confused now..

    Consider the parabola 4ay = x^2, where a > 0 and suppose the tangents at P(ap, ap^2) and Q (2aq, aq^2) intersect at point T.

    Let S (0, a) be the focus of the parabola.

    1) Find the coordinates of T. (assume that the tangents at P is y = px - ap^2).

    2) Show that SP = a(p^2 +1). Suppose P and Q move on the parabola in such a way that SP + SQ = 4a

    3) Show that T is constrained to move on a parabola
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  2. #2
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    Re: Parabola question

    Have you done any work so far? Try typing up as much as you can.
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  3. #3
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    Talking Re: Parabola question

    O, o I solve (1) yay!

    What I did is tangent P is equal to y = px - ap^2
    while tangent Q is equal to y = qx - aq^2

    px - ap^2 = qx - aq^2

    x = (p+q)a

    y = p((p+q)a) - ap^2 = apq

    Im so happy right now..!

    (2) is like this i guess..

    SP + SQ = 4a

    SP + (a(q^2 - 1) = \frac{x^2}{y}

    SP + (a(q^2 - 1) = \frac{ap+aq}{apq}

    SP  = \frac{1}{q} + \frac{1}{a}- (aq^2 - aq)
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  4. #4
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    Re: Parabola question

    For (1), you have it correct. For (2), I am not sure what you are doing. You are supposed to show that SP = a(p^2+1). You can do that by using the distance formula. LaTeX is not working, so plugging in the information to the distance formula:

    [tex]SP = \sqrt{ (2ap - 0)^2 + (ap^2 - a)^2 }[/tex]

    Simplifying, you will get SP = a(p^2+1) as you wanted. Similarly, [tex]SQ = a(q^2+1)[/tex].

    Then, for (3), you suppose that [tex]SP+SQ=4a[/tex], then show that the point T will move along a parabola.

    So, you get [tex]SP+SQ = a(p^2+1) + a(q^2+1) = a(p^2+q^2+2) = 4a[/tex]. Hence, [tex]p^2 + q^2 = 2[/tex].

    From the formula for T you found in part (1), you know [tex]x = a(p+q)[/tex]. Squaring both sides, you get

    [tex]x^2 = a^2(p+q)^2 = a^2(p^2+2pq+q^2) = a^2(p^2+q^2) + 2a(apq) = 2a^2 + 2ay[/tex]

    Solving for y, you get: [tex]y = \dfrac{1}{2a}x^2 - a[/tex] which is the definition for a parabola as desired.
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  5. #5
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    Re: Parabola question

    Thanks SlipEternal
    But im kind of confused in the distance formula,

    SP = \sqrt{ (2ap - 0)^2 + (ap^2 - a)^2 }

    But why do you use 2ap in (2ap - 0)^2 instead of ap ? P's x coordinate is ap right?
    Last edited by Deci; February 8th 2014 at 03:14 AM.
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  6. #6
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    Re: Parabola question

    Quote Originally Posted by Deci View Post
    Thanks SlipEternal
    But im kind of confused in the distance formula,

    SP = \sqrt{ (2ap - 0)^2 + (ap^2 - a)^2 }

    But why do you use 2ap in (2ap - 0)^2 instead of ap ? P's x coordinate is ap right?
    No, it is 2ap. You had a typo. The point (ap,ap^2) is not a point on the parabola. Here is why:

    4ay = x^2 is the formula for the parabola. Plug in ap for x and ap^2 for y: 4a(ap^2) = 4a^2p^2 does not equal a^2p^2. On the other hand, if you change x to 2ap, on the right hand side you get 4a^2p^2, which is the same as the left hand side.
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