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Math Help - Alg. Geom. Proofs

  1. #1
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    Alg. Geom. Proofs



    I'm not very good at thinking about these things... any help is appreciated, it's due 10/27.

    This thread has problem 1.2 in it:

    http://www.mathhelpforum.com/math-he...ns-origin.html
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  2. #2
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by Varitron View Post


    I'm not very good at thinking about these things... any help is appreciated, it's due 10/27.

    This thread has problem 1.2 in it:

    http://www.mathhelpforum.com/math-he...ns-origin.html
    for 1.3..

    let x=r\cos \theta and y = r\sin \theta, thus r^2 = x^2 + y^2

    try changing \sin 3\theta into into something with \cos \theta and \sin \theta only..

    after that, try multiplying r^3 on both sides of r=\sin 3\theta..

    for 1.4..

    let C(x,y) = (x-r)^2 + y^2 - r^2 and let
    D(x,y) = (x-h)^2 + (y-k)^2 - R^2

    solve for I_{(0,0)}(C,D) by taking the cases k=0 and k\neq 0
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  3. #3
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    I got 1.3 actually, and I did exactly that. It worked out.

    For 1.4, I'm still a bit confused. Why can you put (x-r^2)^2 instead of just x^2 for C, and how do we know that D even contains the origin?

    EDIT: We know D contains the origin because the problem says it does! I overlooked that. I'm still having trouble computing I(C,D) at the origin though. Showing some steps would help though. I keep thinking that either r or R need to be 0 otherwise the origin isn't contained. I wound up having something like x(2r-2h) - R^2 on one side, and I'm just very confused.
    Last edited by Varitron; October 26th 2008 at 11:12 PM.
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  4. #4
    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by Varitron View Post
    I got 1.3 actually, and I did exactly that. It worked out.

    For 1.4, I'm still a bit confused. Why can you put (x-r^2)^2 instead of just x^2 for C, and how do we know that D even contains the origin?

    EDIT: We know D contains the origin because the problem says it does! I overlooked that. I'm still having trouble computing I(C,D) at the origin though. Showing some steps would help though. I keep thinking that either r or R need to be 0 otherwise the origin isn't contained. I wound up having something like x(2r-2h) - R^2 on one side, and I'm just very confused.
    r and R are the radii of C and D respectively.. so if either one is 0, then the curve is just a point..

    now, C(x,y) = (x-r)^2 + y^2 - r^2 = x^2 -2rx + y^2
    and D(x,y) = (x-h)^2 + (y-k)^2 - R^2 = x^2 -2hx + h^2 + y^2 - 2ky + k^2 - R^2 (in fact, R^2 = k^2 + h^2 (why?))

    so D(x,y) = x^2 -2hx + y^2 - 2ky

    if the center of D is on the x-axis, that is, k=0, then

    I_{(0,0)}(x^2 -2rx + y^2, x^2 -2hx + y^2) = ...

    if the center is not on the x-axis, i.e., k\neq 0 then

    I_{(0,0)}(x^2 -2rx + y^2, x^2 -2hx + y^2 -2ky) = ...

    hope you can do it from here..
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