x - MathnMind

Your Notes
Homework
Example 2 Use eccentricity to write an equation
Write an equation of a hyperbola with center (21, 3),
vertex (3, 3), and e 5 3.
Solution
Use the form (x2h)2 (y2k)2
} 2 } 5 1. The vertex lies
a2 b2 3 2 (21) 5 4 units from the center, so a 5 4 .
Because e 5 c
}
a 5 3, you know that c
} 5 3, or c 5 12 .
4
So, b2 5 c22a2 5 144 2 16 5 128 .
The equation is
(x 1 1)2
}
16
2 (y 2 3)2
}
128
5 1.
Example 3 Use eccentricity to write a model
An asteroid orbits the sun in an elliptical path with
the sun at one focus. The eccentricity of the orbit
is e 5 0.229 and the length of the major axis is 9.0
astronomical units (A.U). Find an equation of the
orbit. (Assume that the major axis is horizontal.)
Solution
The equation of the orbit has the form x2 y2
} 1 } 5 1. You
a2 b2 know that 2a 5 9.0 , or a 5 4.5 .
e 5 c
}
a , so 0.229 5 c
} , or c ≈ 1.03. For an ellipse,
4.5
c2 = a22 b2 , so b = Ï }
a2 2c2 5 Ï }}
(4.5) 2 2 (1.03) 2 ≈ 4.4
So, an equation for the asteroid’s orbit is
x2 y2 x2 y2
} 1 } 5 1 or } 1 } 5 1, where x and y are
(4.5) 2 (4.4) 2 20 19
measured in A.U.
Checkpoint Complete the following exercise.
3. For another asteroid, the eccentricity of the orbit is
e = 0.459 and the length of the major axis
(horizontal) is 14.8 A.U. Find an equation of the
asteroid’s orbit. x2 y2
} 1 } 5 1
55 43
258 9.7 Focus On Graphing Algebra 2 Notetaking Guide Copyright © Holt McDougal. All rights reserved.