Advanced Geometry
Conic Sections
Lesson 4
Ellipses & Hyperbolas
Ellipses
V
Major
Axis
V
F
C
Minor
Axis
F
V
V
Definition – the set
of all points in a
plane that the sum
of the distances
from two given
points, called the
foci, is constant
Equation
(a² > b²)
Center
Foci
Minor
Axis
a
2
y k


b
2
2
1
y k
a
2
2
x  h


b
2
( h, k )
( h, k )
( h  c, k )
(h, k  c)
equation
yk
vertices
( h  a, k )
equation
xh
xh
(h, k  a)
yk
vertices
(h, k  b)
(h  b, k )
c  a 2  b2
Major
Axis
 x  h
2
2
1
Example:
For the equation of each ellipse or hyperbola, find all information
listed. Then graph.
 x  1   y  2  1
36
9
2
Center:
Foci:
Length of the
major axis:
Length of the
minor axis:
2
Hyperbola
Asymptote
F
C
Asymptote
V
V
F
Transverse
Axis
Conjugate
Axis
Definition – the set
of all points in a
plane that the
absolute value of
the distance from
two given points in
the plane, called the
foci, is constant
Equation of a
Hyperbola
Center
Foci
c  a 2  b2
Vertices
Slopes of the
Asymptotes
Direction of
Opening
 x  h
a
2
2
y k


b
2
2
1
 y k
a
2
2
x  h


b
2
( h, k )
( h  c, k )
(h, k  c)
( h  a, k )
(h, k  a)
2
1
( h, k )
a
b
b
a

left and right
up and down

Example:
For the equation of each ellipse or hyperbola, find all information
listed. Then graph.
 y  1   x  1  1
25
16
2
Center:
Vertices:
Foci:
Slopes of the
asymptotes:
2
Example:
Using the graph below, write the equation for the ellipse or hyperbola.
Example:
Using the graph below, write the equation for the ellipse or hyperbola.
Example:
Write the equation of the ellipse or hyperbola that meets
each set of conditions.
The foci of an ellipse are (-5, 3) and (3, 3) and the
minor axis is 6 units long.
Example:
Write the equation of the ellipse or hyperbola that meets
each set of conditions.
The vertices of a hyperbola are (0,-3) and (0, -8) and the
length of the conjugate axis is 2 6 units long.
Example:
Write each equation in standard form. Determine if it is an
ellipse or a hyperbola.
32  x  1  18  y  4   144  0
2
2