1. Notes 8.3 Conics Sections – The Hyperbola

2. 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.

3. 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.

4. 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.

5. D.) Pictures – By Definition -
P(x, y)
Focus
(x, y)
Focus d2 d1
(-c, 0)
(c, 0)
Vertex(-a, 0)
Vertex(a, 0)

6. Pictures -Expanded- Conjugate Axis Transverse Axis Focus Focus
(0, b)
Focal Axis
(-c, 0)
(c, 0)
(0, -b)
Vertex(-a, 0)
Vertex(a, 0)
Asymptotes are or

7. E.) Standard Form -
Where b2 + a2 = c2.

8. F.) HYPERBOLAS - Center at (0,0)
St. fm..
Focal axis
Foci
Vertices
Semi-Trans.
Semi-Conj.
Pyth. Rel.
Asymptotes

9. G.) HYPERBOLAS - Center at (h, k)
St. fm..
Focal axis
Foci
Vertices
Semi-Trans.
Semi-Conj.
Pyth. Rel.
Asymptotes

10. II.) Examples
A.) Ex. 1- Find the vertices and foci of the following hyperbolas:
Vertices =
Vertices =
Foci =
Foci =

11. B.) Ex. 2- Find an equation in standard form of the hyperbola with
1.) foci (0,±15) and transverse axis of length 8.
2.) Vertices (1, 2) and (1, -8) and conjugate axis of length 6.

12. C.) Ex. 3 - Find the equation of a hyperbola with center at (0, 0), a = 4, e = , and containing a vertical focal axis.

13. III.) Discriminant Test
A.) The second degree equation
is a hyperbola if
a parabola if
an ellipse if
except for degenerate conics

14. B.) Ex. 1 – Identify the following conics:
1.)
2.)
Hyperbola
Ellipse