1. Barnett/Ziegler/Byleen College Algebra, 6th Edition
Chapter Nine
Additional Topics in Analytic Geometry
Copyright © 1999 by the McGraw-Hill Companies, Inc.

2. Conic Sections
Circle Ellipse
Parabola Hyperbola
9-1-80

3. Standard Equations of a Parabola
with Vertex at (0, 0)
1. y2 = 4ax Vertex: (0, 0) Focus: (a, 0) Directrix: x = –a Symmetric with respect
to the x axis Axis the x axis
a < 0 (opens left) a > 0 (opens right)
2. x2 = 4ay Vertex: (0, 0) Focus: (0, a) Directrix: y = –a Symmetric with respect
to the y axis Axis the y axis
a < 0 (opens down) a > 0 (opens up)
9-1-81

4. Standard Equations of an Ellipse with Center at (0, 0)
[Note: Both graphs are symmetric with respect to the x axis,
y axis, and origin. Also, the major axis is always longer than the minor axis.]
9-2-82

5. Standard Equations of a Hyperbola
with Center at (0, 0) x 2 y 2 1. +
= 1 a 2 b 2 x
intercepts: ± a
(vertices) y
intercepts: none
Foci: F' (– c
, 0) F ( c
, 0) c 2 = a 2 + b 2
Transverse axis length = 2 a
Conjugate axis length = 2 b y 2 x 2 2. –
= 1 a 2 b 2 x
intercepts: none y
intercepts: ± a
(vertices)
Foci: F'
(0, – c ) F
(0, c ) c 2 = a 2 + b 2
Transverse axis length = 2 a
Conjugate axis length = 2 b
[Note: Both graphs are symmetric with respect
to the x axis, y axis, and origin.]
9-3-83

6. Standard Equations for
Translated Conics—I
(x – h)2 = 4a(y – k)
Vertex (h, k)
Focus (h, k + a)
a > 0 opens up
a < 0 opens down
Parabolas
Circles
(x – h)2 + (y – k)2 = r2
Center (h, k) Radius r
(y – k)2 = 4a(x – h)
Vertex (h, k)
Focus (h + a, k)
a < 0 opens left
a > 0 opens right
9-4-84

7. Standard Equations for Translated Conics—II
Ellipses

8. Standard Equations for Translated Conics—II
Hyperbolas