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Thread: True or false?

  1. #1
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    True or false?

    $\displaystyle x^2+y^2-2x-4y+6=0$ is the equation of a circle.

    I answered true, but the answer in the book is false. So who is correct? This equation satisfies the general equation of a circle $\displaystyle x^2+y^2+2gx+2fy+c=0$
    Thanks
    Last edited by arze; Oct 5th 2009 at 05:04 PM. Reason: latex
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  2. #2
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    Quote Originally Posted by arze View Post
    $\displaystyle x^2+y^2-2x-4y+6=0$ is the equation of a circle.

    I answered true, but the answer in the book is false. So who is correct? This equation satisfies the general equation of a circle $\displaystyle x^2+y^2+2gx+2fy+c=0$
    Thanks
    $\displaystyle x^2 - 2x + y^2 - 4y = -6$

    complete the square for x and y ...

    $\displaystyle x^2 - 2x + 1 + y^2 - 4y + 4 = -6 + 1 + 4$

    $\displaystyle (x-1)^2 + (y-2)^2 = -1$

    since the general circle equation is

    $\displaystyle (x-h)^2 + (y-k)^2 = r^2$


    $\displaystyle r^2 = -1$ ?
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  3. #3
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    Quote Originally Posted by arze View Post
    $\displaystyle x^2+y^2-2x-4y+6=0$ is the equation of a circle.

    I answered true, but the answer in the book is false. So who is correct? This equation satisfies the general equation of a circle $\displaystyle x^2+y^2+2gx+2fy+c=0$
    Thanks
    Actually, the "general equation" of a circle is of the form

    $\displaystyle (x - h)^2 + (y - k)^2 = r^2$

    which is a circle of radius $\displaystyle r$, centred at $\displaystyle (h, k)$.


    I would try to transform the equation you have been given into this form.

    $\displaystyle x^2 + y^2 - 2x - 4y + 6 = 0$

    $\displaystyle x^2 - 2x + y^2 - 4y = -6$

    $\displaystyle x^2 - 2x + (-1)^2 + y^2 - 4y + (-2)^2 = -6 + (-1)^2 + (-2)^2$

    $\displaystyle (x - 1)^2 + (y - 2)^2 = -1$.


    Is $\displaystyle r^2 = -1$ possible?
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    Ah!!! How could I have missed that. Thanks!
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