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 Vertex (-a, 0) (a, 0) Pictures -Expanded- Conjugate Axis Transverse Axis Focus Focus (0, b) Focal Axis (-c, 0) (c, 0) (0, -b) Vertex Vertex (-a, 0) (a, 0) a b Asymptotes areyx or yx b a E.) Standard Form - xy22 yx 22 1 or 1 ab22 ab 22 Where b2 + a2 = c2. F.) HYPERBOLAS - Center at (0,0) xy22 yx22 St. fm.. 1 22 1 ab22 ab Focal axis x axis yaxis Foci c,0 0, c Vertices a,0 0, a Semi-Trans. a a Semi-Conj. b b 222 Pyth. Rel. cab222 cab b a yx yx Asymptotes a b G.) HYPERBOLAS - Center at (h, k) 22 22 xh yk yk xh St. fm.. 22 1 1 ab ab22 Focal axis yk xh Foci hck , hk, c Vertices hak , hk, a Semi-Trans. a a Semi-Conj. b b 222 Pyth. Rel. cab222 cab b a yxhk yxhk Asymptotes a b II.) Examples A.) Ex. 1- Find the vertices and foci of the following hyperbolas: 22 22 yx32 1.) 3xy 4 12 2.) 1 94 xy22 1 43 Vertices = 2,0 Vertices = 2,6 and ( 2,0) 7,0 Foci = 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. yx22 1 16 209 2.) Vertices (1, 2) and (1, -8) and conjugate axis of length 6. 22 yx31 1 25 9 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 cc3 e 222 a 24 cab 36 16 b2 c 6 20 b2 yx22 1 16 20 III.) Discriminant Test A.) The second degree equation Ax22 Bxy Cy Dx Ey F 0 is a hyperbola if BAC2 40 a parabola if BAC2 40 an ellipse if BAC2 40 except for degenerate conics B.) Ex. 1 – Identify the following conics: 1.) 2xy22 3 12 x 24 y 60 0 BAC2 404230 Hyperbola 2.) 10xxyyxy22 8 6 8 5 30 0 BAC2 46441060 Ellipse.