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) (c, 0) Vertex(-a, 0) Vertex(a, 0)

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

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

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

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

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

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.

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

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

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