Intermediate Algebra (11th edition)

When a ray of light or sound emanating from one focus of an ellipse bounces off

the ellipse, it passes through the other focus. See FIGURE 14. As mentioned in the

chapter introduction, this reflecting property is responsible for whispering galleries.

John Quincy Adams was able to listenin on his opponents’ conversations because his

desk was positioned at one of the foci beneath the ellipsoidal ceiling and his oppo-

nents were located across the room at the other focus.

Elliptical bicycle gears are designed to respond to the legs’ natural strengths and

weaknesses. At the top and bottom of the powerstroke, where the legs have the least

leverage, the gear offers little resistance, but as the gear rotates, the resistance in-

creases. This allows the legs to apply more power where it is most naturally available.

See FIGURE 15.

OBJECTIVE 4 Graph ellipses.

Graphing Ellipses

Graph each ellipse.

(a)

Here, so and the x-intercepts are and Similarly,

so and the y-intercepts are and Plotting the inter-

cepts and sketching the ellipse through them gives the graph in FIGURE 16.

b^2 =36, b=6, 1 0, 6 2 1 0, - 62.

a^2 =49, a= 7, 1 7, 0 2 1 - 7, 0 2.

x^2

49

+

y^2

36

= 1

EXAMPLE 5

SECTION 11.2 The Circle and the Ellipse 645

y

x

0

Reflecting property

of an ellipse

FIGURE 14

FIGURE 15

(b)

The x-intercepts are and and the y-intercepts are and

1 0, - 112 .Join these with the smooth curve of an ellipse. See FIGURE 17.

1 6, 0 2 1 - 6, 0 2 , 1 0, 11 2

x^2

36

+

y^2

121

= 1

0

x

y

6

• 77
  
  
  • 6 xx^22
    4949
  
  
  
  

yy^22

3636 11

FIGURE 16

0

x

y

• 11

11

• 662

xx^22

3636

yy^22

121121 11

FIGURE 17 NOW TRY

NOW TRY

EXERCISE 5

Graph.

x^2

16

+

y^2

25

= 1

NOW TRY ANSWER
5.

x

y

–4^04

–5

5

+= 1 25 y

2

16

x^2