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Math Help - Simultaneous Equation

  1. #1
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    Simultaneous Equation

    Hiya,

    I need to find the range of values of k which give distinct roots, given the following 2 equations:

    kx + y = 3
    x^2 + y^2 = 5

    Thanks guys.

    Kermola
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  2. #2
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    Re: Simultaneous Equation

    Simultaneous Equation-circle-tangent.png

    The angles B and B' are 90, but angle BAB' is not 90. You can find \cos(\angle AOB'), from where you can find the slope of AB' (recall that \tan(\alpha)\tan(\alpha+\pi/2)=-1).
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  3. #3
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    Re: Simultaneous Equation

    Thanks for that.
    Any way to solve it without trigonometry and without drawing out the graphs? i.e. solve it algebraically?
    Thanks
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  4. #4
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    Re: Simultaneous Equation

    You can express y from the first equation and substitute it into the second one. You'll get a quadratic equation in x with coefficients depending on k. Determine for which k the discriminant is positive.
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  5. #5
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    Re: Simultaneous Equation

    Oh I've managed to figure it out!
    Correct me if I'm wrong :

    from (1) - y = 3 - kx
    therefore x^2 + (3-kx)^2 = 5
    x^2 + 9 - 6kx + 9 = 5
    (k^2 + 1)x^2 - 6kx + 4 = 0
    using b2 - 4ac to find real roots
    (-6k)^2 - 16k^2 - 16 > 0
    36k^2 - 16k^2 - 16 > 0
    20k^2 > 16
    k^2 > 4/5
    -2/(5^(1/2)) < k > 2/(5^(1/2))


    Hmm.. what program do people use to write proper it in proper mathematical form to post on these forums? That'd be helpful for the future.

    Kermola
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  6. #6
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    Re: Simultaneous Equation

    Ha-ha, I was early by 5 seconds!

    Quote Originally Posted by kermola View Post
    Oh I've managed to figure it out!
    Correct me if I'm wrong :

    from (1) - y = 3 - kx
    therefore x^2 + (3-kx)^2 = 5
    x^2 + 9 - 6kx + 9 = 5
    (k^2 + 1)x^2 - 6kx + 4 = 0
    using b2 - 4ac to find real roots
    (-6k)^2 - 16k^2 - 16 > 0
    36k^2 - 16k^2 - 16 > 0
    20k^2 > 16
    k^2 > 4/5
    -2/(5^(1/2)) < k < 2/(5^(1/2))
    The last line is incorrect.

    Quote Originally Posted by kermola View Post
    Hmm.. what program do people use to write proper it in proper mathematical form to post on these forums? That'd be helpful for the future.
    See the LaTeX Help subforum. Wrap LaTeX code in [TEX]...[/TEX] tags.
    Thanks from kermola
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  7. #7
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    Re: Simultaneous Equation

    I've since editted my post - we keep missing each other!

    Thanks for the latex pointer - i'll check it out.
    Is the editted post correct?
    Thanks so much for help.

    Kerm
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  8. #8
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    Re: Simultaneous Equation

    Quote Originally Posted by kermola View Post
    -2/(5^(1/2)) < k > 2/(5^(1/2))
    It should be k > 2/(5^(1/2)) or k < -2/(5^(1/2)). When you want to abbreviate k > x and k < y, it's OK to write x < k < y. However, it is not customary to abbreviate k < x or k > y. If you write y < k < x, it would be confused with k < x and k > y.
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  9. #9
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    Re: Simultaneous Equation

    I was just being a bit lazy when I edited the post. Thanks a lot for the help!

    K
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