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Math Help - A Circle Question.

  1. #1
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    A Circle Question.

    Hi,
    Can anyone give me a hand with the following problem please?

    For what range of values of k does: x^2+y^2+4kx-2ky-k-2+0
    represent a circle?

    I have tried using the radius formula :-

    Sqrrt (-2k)^2 + (k)^2 -(k-1)

    Sqrrt 4k^2 + k^2 -k+1

    Sqrrt 5k^2 -k+1......
    Thanks
    Cromlix
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  2. #2
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    Quote Originally Posted by cromlix View Post
    Hi,
    Can anyone give me a hand with the following problem please?

    For what range of values of k does: x^2+y^2+4kx-2ky-k-2+0
    represent a circle?


    I have tried using the radius formula :-

    Sqrrt (-2k)^2 + (k)^2 -(k-1)

    Sqrrt 4k^2 + k^2 -k+1

    Sqrrt 5k^2 -k+1......
    Thanks
    Cromlix
    I guess the equation you wanted to write was x^2+y^2+4kx-2ky-k-2=0
    the formula for the radius is \sqrt{g^2+f^2-c}
    i guess you have made a calculation mistake there.
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  3. #3
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    Quote Originally Posted by cromlix View Post
    Hi,
    Can anyone give me a hand with the following problem please?

    For what range of values of k does: x^2+y^2+4kx-2ky-k-2+0
    represent a circle?

    I have tried using the radius formula :-

    Sqrrt (-2k)^2 + (k)^2 -(k-1)

    Sqrrt 4k^2 + k^2 -k+1

    Sqrrt 5k^2 -k+1......
    Thanks
    Cromlix
    \displaystyle x^2 + y^2 + 4k\,x - 2k\,y -k- 2 = 0

    \displaystyle x^2 + 4k\,x + (2k)^2 + y^2 - 2k\,y + (-k)^2-k - 2 = (2k)^2 + (-k)^2

    \displaystyle (x + 2k)^2 + (y - k)^2 -k- 2 = 4k^2 + k^2

    \displaystyle (x + 2k)^2 + (y - k)^2 = 5k^2 + k+2.


    Now all that's left is to show the values for \displaystyle k for which the RHS is nonnegative (complete the square).
    Last edited by Prove It; February 12th 2011 at 08:53 AM.
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  4. #4
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    Although Prove_It missed a k as in (x+2k)^2+(y-k)^2=k+2+5k^2 it does not change his answer.
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  5. #5
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    Quote Originally Posted by Plato View Post
    Although Prove_It missed a k as in (x+2k)^2+(y-k)^2=k+2+5k^2 it does not change his answer.
    Fixed...
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  6. #6
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    Thanks for your work.
    I have completed the square (x+2k)^2 + (y-k)^2= 5k^2+k+2

    5( k^2 + 0.2k) +2
    5( k^2 +0.2k +0.01) -0.05 + 2
    5(k + 0.1)^2 +1.95

    How do I find the range from this?

    Thank you
    Cromlix
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  7. #7
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    All you had to do was show that this is always nonnegative.

    Can a square ever give a negative value?
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