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Math Help - Parametric Problem Involving Elipses

  1. #1
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    Parametric Problem Involving Elipses

    I was set the following problem as part of a homework assignment:

    P is a point on the ellipse with parametric equations x=5cost, y=3sint for 0 <= t < 2pi, and F and G are the points (-4,0) and (4,0) respectively. Prove that:

    a) FP = 5 + 4cost
    b) FP + PG = 10

    Let the normal at P make angles θ and Φ with FP and GP respectively. Prove that:

    c) tan θ = (4sint)/3
    d) θ = Φ

    Whilst I found parts (a) and (b) trivial to prove, I could not work out a way to find the tangent of θ for part (c). After taking the problem home, my maths teacher could also not solve the problem! Does anybody have any suggestions?

    Thanks in advance.
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  2. #2
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    The equation of the ellipse is
    x^2/a^2 + y^2/b^2 = 1.
    Substitute the values of x and y in the above equation. Guess the values of a and b so that equation equals to one.
    Using the relation e^2 = 1 - b^2/a^2, find the eccentricity e of the ellipse.
    Co-ordinates of foci are (-ae, 0) and (ae, 0). So you can verify that F and G are fosi.
    If PM' is the distance of P from the diretrix on the side of F, then
    FP = e*PM' = e*(a/e + x).
    Complete this part first.
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  3. #3
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    sa-ri-ga-ma - Thanks for your reply but the maths you have suggested to use is not part of my course and something I haven't learnt yet. There surely must be a more simple method... This question comes from a british A Level textbook.
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  4. #4
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    Quote Originally Posted by StaryNight View Post
    sa-ri-ga-ma - Thanks for your reply but the maths you have suggested to use is not part of my course and something I haven't learnt yet. There surely must be a more simple method... This question comes from a british A Level textbook.
    What are the things you know about ellipse at the present level of understanding? Let us see if there is any way to proceed.
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  5. #5
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    Quote Originally Posted by sa-ri-ga-ma View Post
    What are the things you know about ellipse at the present level of understanding? Let us see if there is any way to proceed.
    We actually haven't studied elipses at all - only circles. To put this question in context it is from an excercise about parametric equations and finding the equations of normals/tangents to curves.
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  6. #6
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    Quote Originally Posted by StaryNight View Post
    We actually haven't studied elipses at all - only circles. To put this question in context it is from an excercise about parametric equations and finding the equations of normals/tangents to curves.
    In that case, you cannot solve the given problem by any simple method. For (c) and (d) part, you must know all about the ellipse.
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