Analytic Geometry Formulas

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

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)
           
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: