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