1. Circle
(x - h)2 + (y - k)2 = r2
center (h, k)
radius r
2. Parabola
(x - h)2 = 4p (y - k)
opens up
vertex (h, k)
focus (h, k + p)
directrix y = k - p
(x - h)2 = -4p (y - k)
opens down
vertex (h, k)
focus (h, k - p)
directrix y = k + p
(y - k)2 = 4p (x - h)
opens right
vertex (h, k)
focus (h + p, k)
directrix x = h - p
(y - k)2 = -4p (x - h)
opens left
vertex (h. k)
focus (h - p, k)
directrix x = h + p
3. Ellipse
center (h, k)
a2 - b2 = c2 or
a2 = b2 + c2
foci (h - c, k), (h + c, k)
sum of distances to foci = 2a
major axis is parallel to x-axis = 2a
minor axis is parallel to y-axis = 2b
eccentricity = c / a
vertices (h + a, k), (h - a, k),
(h, k + b), (h, k - b)
center (h, k)
a2 - b2 = c2 or
a2 = b2 + c2
foci (h, k+c), (h, k-c)
sum of distances to foci = 2a
major axis is parallel to x-axis = 2a
minor axis is parallel to y-axis = 2b
eccentricity = c / a
vertices (h + b, k), (h - b, k),
(h, k + a), (h, k - a)
4. Hyperbola
center is (h, k)
c2 = a2 + b2
vertices (h + a, k), (h - a, k)
foci (h + c, k), (h - c, k)
asymptotes:
center is (h, k)
c2 = a2 + b2
vertices (h, k + a), (h, k - a)
foci (h, k + c), (h, k - c)
asymptotes:
