Conic Sections Formulas Cheat Sheet
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1Conic Sections Formulas
Parabola
Vertical Axis
Horizontal axis
2
2
equation
(x-h)
=4p(y-k)
(y-k)
=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
p>0 then rignt; p<0 then
down
left
Ellipse
Vertical Major Axis
Horizontal Major axis
equation
2
2
2
2
x h
y k
x h
y k
1
1
2
2
2
2
b
a
a
b
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,
2
2
2
a
=b
+c
and c
Eccentricity of an ellipse
e=(c/a)
Hyperbola
Vertical Transverse Axis
Horizontal Transverse axis
equation
2
2
2
2
y k
x h
x h
y k
1
1
2
2
2
2
a
b
a
b
center
(h,k)
(h,k)
Vertices
(h,k±a)
(h±a,k)
Foci
(h,k±c)
(h±c,k)
Assymptote equation
a
b
y k
x h
y k
x h
b
a
Equation relating a, b,
2
2
2
c
=a
+b
and c
Classifying conic
Circles
Parabola
Ellipse
Hyperbola
sections
AC=0, Both
AC>0
AC<0
2
2
Ax
+Cy
+Dx+Ey+F=0
A=C
are not 0
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