Conic Sections

Parabola with vertex (h, k)

Opens up or down

(x - h)^{2} = 4 p (y - k)

focus (h, k + p), directrix y = k − p

Opens left or right

(y - k)^{2} = 4 p (x - h)

focus (h + p, k), directrix x = h − p

Wide (horizontal major axis)

\frac{(x - h)^{2}}{a^{2}} + \frac{(y - k)^{2}}{b^{2}} = 1

foci (h ± c, k), vertices (h ± a, k)

Tall (vertical major axis)

\frac{(x - h)^{2}}{b^{2}} + \frac{(y - k)^{2}}{a^{2}} = 1

foci (h, k ± c), vertices (h, k ± a)

Area

A = π a b

Eccentricity

e = \frac{c}{a}

0 < e < 1

Opens left and right

\frac{(x - h)^{2}}{a^{2}} - \frac{(y - k)^{2}}{b^{2}} = 1

foci (h ± c, k), asymptotes y − k = ±(b/a)(x − h)

Opens up and down

\frac{(y - k)^{2}}{a^{2}} - \frac{(x - h)^{2}}{b^{2}} = 1

foci (h, k ± c), asymptotes y − k = ±(a/b)(x − h)

Ellipse or circle

B^{2} - 4 A C < 0

Parabola

B^{2} - 4 A C = 0

Hyperbola

B^{2} - 4 A C > 0

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