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Pre-calculus and algebra course notes on conic section
Typology: Cheat Sheet
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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
2 2
2 2
x h y k 1 b a
2 2
2 2
x h y k 1 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,
and c
a
2 =b
2 +c
2
Eccentricity of an ellipse e=(c/a)
Hyperbola
Vertical Transverse Axis Horizontal Transverse axis
equation
2 2
2 2
y k x h 1 a b
2 2
2 2
x h y k 1 a b
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
c
2 =a
2 +b
2
Classifying conic sections
Circles Parabola Ellipse Hyperbola
Ax
2 +Cy
2 +Dx+Ey+F=0 A=C
AC=0, Both
are not 0