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Conic Section Formula Sheet with Parabola, Ellipse and Hyperbola, Cheat Sheet of Algebra

Pre-calculus and algebra course notes on conic section

Typology: Cheat Sheet

2020/2021

Uploaded on 04/27/2021

ekavaria
ekavaria 🇺🇸

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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
Circles
Parabola
Ellipse
Hyperbola
A=C
AC=0, Both
are not 0
AC>0
AC<0

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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

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

AC>0 AC<