Notes 8.3
Conics Sections –
The Hyperbola
I. Introduction
A.) The set of all points in a plane
whose distances from two fixed
points(foci) in the plane have a
constant difference.
1.) The fixed points are the FOCI.
2.) The line through the foci is the
FOCAL AXIS.
3.) The CENTER is ½ way between
the foci and/or the vertices.
B.) Forming a Hyperbola - When a plane
intersects a double-napped cone and is
perpendicular to the base of the cone, a
hyperbola is formed.
C.) More Terms
1.) A CHORD connects two points of a
hyperbola.
2.) The TRANSVERSE AXIS is the chord
connecting the vertices. It’s length is equal to
2a, while the semi-transverse axis has a
length of a.
3.) The CONJUGATE AXIS is the line
segment perpendicular to the focal axis. It’s
length is equal to 2b, while the semi-conjugate
axis has a length of b.
D.) Pictures – By Definition P(x, y)
Focus
(x, y)
Focus
d2
d1
(-c, 0)
Vertex
(-a, 0)
(c, 0)
Vertex
(a, 0)
Pictures -ExpandedConjugate Axis
Transverse
Axis
Focus
(0, b)
Focus
(-c, 0)
Focal Axis
(c, 0)
(0, -b)
Vertex
(-a, 0)
Vertex
(a, 0)
a
b
Asymptotes are y   x or y   x
b
a
E.) Standard Form -
2
2
x
y
 2  1 or
2
a
b
Where b2 + a2 = c2.
2
2
y
x
 2 1
2
a b
F.) HYPERBOLAS - Center at (0,0)
St. fm..
Focal axis
Foci
Vertices
y2 x2
 2 1
2
a b
x2 y 2
 2 1
2
a b
x  axis
y  axis
  c, 0 
  a, 0 
 0, c 
 0,  a 
Semi-Trans.
a
a
Semi-Conj.
b
b
Pyth. Rel.
c  a b
Asymptotes
2
2
b
y x
a
2
c  a b
2
2
a
y x
b
2
G.) HYPERBOLAS - Center at (h, k)
 x  h   y  k   1
a2
b2
2
St. fm..
yk
Focal axis
Foci
Vertices
2
 y  k    x  h  1
a2
b2
2
2
xh
 h  c, k 
 h  a, k 
 h, k  c 
 h, k  a 
Semi-Trans.
a
a
Semi-Conj.
b
b
Pyth. Rel.
Asymptotes
c  a b
b
y    x  h  k
2
2
a
2
c  a b
2
2
2
a
y    x  h  k
b
II.) Examples
A.) Ex. 1- Find the vertices and foci of the
following hyperbolas:
1.) 3x  4 y  12
2
2
2.)
 y  3
9
x2 y2

1
4
3
Vertices =
Foci =

 2, 0 
7, 0

2
x  2


4
2
1
Vertices =  2, 6  and (2, 0)

Foci = 2,3  13

B.) Ex. 2- Find an equation in standard form of
the hyperbola with
1.) foci (0,±15) and transverse axis of
length 8.
2
2
y
x

1
16 209
2.) Vertices (1, 2) and (1, -8) and
conjugate axis of length 6.
 y  3
25
2
x  1


9
2
1
C.) Ex. 3 - Find the equation of a hyperbola with center at
(0, 0), a = 4, e = 3 , and containing a vertical focal axis.
2
c 3 c
e  
a 2 4
c  a b
2
2
36  16  b
c6
20  b
2
2
y
x

1
16 20
2
2
2
III.) Discriminant Test
A.) The second degree equation
Ax  Bxy  Cy  Dx  Ey  F  0
2
is
2
a hyperbola if B 2  4 AC  0
a parabola if
B 2  4 AC  0
an ellipse if
B 2  4 AC  0
except for degenerate conics
B.) Ex. 1 – Identify the following conics:
1.)
2 x  3 y  12 x  24 y  60  0
2
2
B 2  4 AC  0  4  2  3  0
Hyperbola
2.)
10 x  8 xy  6 y  8 x  5 y  30  0
2
2
B 2  4 AC  64  4 10  6   0
Ellipse