Thread: parabola
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Old December 3rd, 2008, 05:44 PM
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skeeter skeeter is offline
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y = x^2 - 5x + 4

roots ...

x^2 - 5x + 4 = 0

(x - 1)(x - 4) = 0

see the roots?

to get the focus and directrix, need to find the vertex coordinates ...

x = \frac{-b}{2a} = \frac{5}{2}

y = \left(\frac{5}{2}\right)^2 - 5\left(\frac{5}{2}\right) + 4

y = -\frac{9}{4}

focal length, d = \frac{1}{4a} = \frac{1}{4}

focus is at \left(\frac{5}{2} , -\frac{9}{4} + \frac{1}{4}\right) = \left(\frac{5}{2} , -2\right)

directrix is y = -\frac{9}{4} - \frac{1}{4} = -\frac{5}{2}


general form for a hyperbola centered at the origin ...

\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1

asymptotes are y = \frac{b}{a} x and y = -\frac{b}{a} x

focal length c = \sqrt{a^2 + b^2} ... focal points are (-c,0) and (c,0)
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