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Notes 8.3

Conics Sections – The I. Introduction

A.) The of all points in a whose distances from two fixed points(foci) in the plane have a constant difference. 1.) The fixed points are the FOCI. 2.) The 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 and is to the 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 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) d2 d1

(-c, 0) (c, 0)

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 areyx or yx b a E.) Standard Form -

xy22 yx 22  1 or 1 ab22 ab 22

Where b2 + a2 = c2. F.) - 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 yx32 1.) 3xy 4 12 2.)  1 94 xy22 1 43

Vertices = 2,0 Vertices = 2,6 and ( 2,0)

Foci =  7,0 Foci = 2,3 13 B.) Ex. 2- Find an 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 yx31  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.) Test A.) The second degree equation Ax22 Bxy Cy Dx Ey F 0 is a hyperbola if BAC2  40 a if BAC2  40 an 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