FORMULAS FOR CHAPTER 6
Conic Sections with Center at the Origin:
CIRCLES
Equation
x2  y 2  r 2
ELLIPSES
Equation
2
2
x2 y2

 1; where a  b
b2 a 2
x
y
 2  1; where a  b
2
a
b
Vertices
  a, 0 
and  a, 0 
 0,  a 
and  0, a 
Foci
 c, 0 
and  c, 0 
 0,  c 
and  0, c 
where c  a 2  b 2
HYPERBOLAS
Equation
2
2
y2 x2

1
a 2 b2
x
y
 2 1
2
a
b
Vertices
  a, 0 
and  a, 0 
 0,  a 
and  0, a 
Foci
 c, 0 
and  c, 0 
 0,  c 
and  0, c 
where c  a 2  b 2
Asymptotes
y
b
x
a
y
a
x
b
PARABOLAS with Vertex at the Origin
Equation
y 2  4 px
x 2  4 py
Focus
 p, 0 
 0, p 
Directrix
x  p
y  p
Formulas for Ch. 6 (cont.)
Conic Sections with Center (h, k):
CIRCLES
 x  h   y  k 
Equation
2
2
 r2
ELLIPSES
Equation
 x  h
a2
2
y k

b2
2
 1; where a  b
 x  h
b2
2
y k

a2
2
 1; where a  b
Vertices
 h  a, k 
and  h  a, k 
 h, k  a 
and  h, k  a 
Foci
 h  c, k 
and  h  c, k 
 h, k  c 
and  h, k  c 
where c  a 2  b 2
HYPERBOLAS
 x  h
Equation
a2
2
y k

b2
y k
2
1
 x  h

2
a2
b2
2
1
Vertices
 h  a, k 
and  h  a, k 
 h, k  a 
and  h, k  a 
Foci
 h  c, k 
and  h  c, k 
 h, k  c 
and  h, k  c 
where c  a 2  b 2
Asymptotes
yk  
b
 x  h
a
yk  
a
 x  h
b
PARABOLAS with Vertex at (h, k)
Equation
Focus
Directrix
 y k
2
 4 p  x  h
 h  p, k 
x  h p
 x  h
2
 4p y  k
 h, k  p 
yk p