Conic Sections Formulas
Parabola
Vertical Axis
Horizontal axis
equation
(x-h)2=4p(y-k)
(y-k)2=4p(x-h)
Axis of symmetry
x=h
y=k
Vertex
(h,k)
(h,k)
Focus
(h,k+p)
(h+p,k)
Directrix
y=k-p
x=h-p
Direction of opening
p>0 then up; p<0 then
down
p>0 then rignt; p<0 then
left
Ellipse
Vertical Major Axis
Horizontal Major axis
equation
22
22
x h y k 1
ba
22
22
x h y k 1
ab
center
(h,k)
(h,k)
Vertices
(h,k±a)
(h±a,k)
Foci
(h,k±c)
(h±c,k)
Major axis equation
2a=length of major axis
Minor axis equation
2b=length of minor axis
Equation that relates a, b,
and c
a2=b2+c2
Eccentricity of an ellipse
e=(c/a)
Hyperbola
Vertical Transverse Axis
Horizontal Transverse axis
equation
22
22
y k x h 1
ab
22
22
x h y k 1
ab
center
(h,k)
(h,k)
Vertices
(h,k±a)
(h±a,k)
Foci
(h,k±c)
(h±c,k)
Assymptote equation
a
y k x h
b
b
y k x h
a
Equation relating a, b,
and c
c2=a2+b2
Classifying conic
sections
Circles
Parabola
Ellipse
Hyperbola
Ax2+Cy2+Dx+Ey+F=0
A=C
AC=0, Both
are not 0
AC>0
AC<0
