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Thread: Conic Sections

  1. #1
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    Conic Sections

    Hey guys, I need some help on Conic Sections. Specifically how to rewrite equations for hyperbola and elipses in standard form using the completeing the square method.

    Here are two example problems from my math book that i have been having trouble with, i factor them using the completeting the square method but the im not sure how to get them into the standard form.

    $\displaystyle 9x^2-2y^2+18=0$
    I know this one is a hyperbola.

    $\displaystyle 2x^2+2y^2-10x-18y=1$
    This one is a parabola i believe.
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  2. #2
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    1.
    $\displaystyle 9x^2 - 2y^2 + 18 = 0$

    $\displaystyle 9x^2 - 2y^2 = -18$

    $\displaystyle \frac{-x^2}{2} + \frac{y^2}{9} = 1$.

    No need to complete the square.


    2.
    $\displaystyle 2x^2 + 2y^2 - 10x - 18y = 1$

    $\displaystyle x^2 - 5x + y^2 - 9y = \frac{1}{2}$

    $\displaystyle x^2 - 5x + \left(-\frac{5}{2}\right)^2 + y^2 - 9y + \left(-\frac{9}{2}\right)^2 = \frac{1}{2} + \left(-\frac{5}{2}\right)^2 + \left(-\frac{9}{2}\right)^2$

    $\displaystyle \left(x - \frac{5}{2}\right)^2 + \left(y - \frac{9}{2}\right)^2 = \frac{1}{2} + \frac{25}{4} + \frac{81}{4}$

    $\displaystyle \left(x - \frac{5}{2}\right)^2 + \left(y - \frac{9}{2}\right)^2 = 27$

    $\displaystyle \left(x - \frac{5}{2}\right)^2 + \left(y - \frac{9}{2}\right)^2 = \left(3\sqrt{3}\right)^2$.

    So this is a circle.
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