College Algebra 9th txtbk

SECTION 6–2 Ellipse 397
Z THEOREM 1 Standard Equations of an Ellipse with Center at (0, 0)
x 2
1. a b 0
a 2 y2
b 2 1
x intercepts: a (vertices)
y intercepts: b
Foci: F (c, 0), F (c, 0)
c 2 a 2 b 2
a
b
y
F
F
c 0
c a
x
Major axis length 2a
Minor axis length 2b
x 2
2. a b 0
b 2 y2
a 2 1
x intercepts: b
y intercepts: a (vertices)
Foci: F(0, c), F (0, c)
b
a
c
y
F
c 2 a 2 b 2
Major axis length 2a
Minor axis length 2b
b
0 b
x
[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.]
c
a
F
ZZZ EXPLORE-DISCUSS 1
The line through a focus F of an ellipse that is perpendicular to the major axis
intersects the ellipse in two points G and H. For each of the two standard equations
of an ellipse with center (0, 0), find an expression in terms of a and b for
the distance from G to H.
EXAMPLE 1 Graphing an Ellipse
Find the coordinates of the foci, find the lengths of the major and minor axes, and graph
the following equation:
9x 2 16y 2 144
SOLUTION First, write the equation in standard form by dividing both sides by 144 and determine a and b:
9x 2 16y 2 144
9x 2
144 16y2
144 144
144
x 2
16 y2
9 1
*
Divide both sides by 144.
Simplify.
a 4 and b 3
*Throughout the book, dashed boxes—called think boxes—are used to represent steps that may be performed
mentally.