Practice and Problem-Solving Exercises

A Practice

See Problem 1.

Write an equation of an ellipse in standard form with center at the origin and with the given vertex and co-vertex listed respectively.

7. (4, 0), (0, 3)
8. (0, 1), (2, 0)
9. (3, 0), ( 0 , − 1 ) open 0 comma negative 1 close
10. (0, 6), (1, 0)
11. ( 0 , − 7 ) , open 0 comma negative 7 close comma  (4, 0)
12. ( − 6 , 0 ) , open negative 6 comma 0 close comma  (0, 5)
13. ( − 9 , 0 ) , ( 0 , − 2 ) open negative 9 comma 0 close comma open 0 comma negative 2 close
14. (0, 5), ( − 3 , 0 ) open negative 3 comma 0 close

See Problem 2.

Find the foci for each equation of an ellipse. Then graph the ellipse.

15. x 2 4 + y 2 9 = 1 fraction x squared , over 4 end fraction , plus , fraction y squared , over 9 end fraction , equals 1
16. x 2 9 + y 2 25 = 1 fraction x squared , over 9 end fraction , plus , fraction y squared , over 25 end fraction , equals 1
17. x 2 81 + y 2 49 = 1 fraction x squared , over 81 end fraction , plus , fraction y squared , over 49 end fraction , equals 1
18. x 2 25 + y 2 16 = 1 fraction x squared , over 25 end fraction , plus , fraction y squared , over 16 end fraction , equals 1
19. x 2 64 + y 2 100 = 1 fraction x squared , over 64 end fraction , plus . fraction y squared , over 100 end fraction . equals 1
20. 3 x 2 + y 2 = 9 3 x squared , plus . y squared , equals 9
21. x 2 + 4 y 2 = 16 x squared , plus 4 , y squared , equals 16
22. x 2 225 + y 2 144 = 1 fraction x squared , over 225 end fraction . plus . fraction y squared , over 144 end fraction . equals 1

See Problem 3.

Find the distance between the foci of an ellipse. The lengths of the major and minor axes are listed respectively.

23. 40 and 24
24. 30 and 18
25. 10 and 8
26. 16 and 10
27. 20 and 16
28. 18 and 14
29. 36 and 12
30. 8 and 6

See Problem 4.

Write an equation of an ellipse for the given foci and co-vertices.

31. foci ( ± 6 , 0 ) , open plus minus 6 comma 0 close comma  co-vertices ( 0 , ± 8 ) open 0 comma plus minus 8 close
32. foci ( 0 , ± 8 ) , open 0 comma plus minus 8 close comma  co-vertices ( ± 8 , 0 ) open plus minus 8 comma 0 close
33. foci ( ± 5 , 0 ) , open plus minus 5 comma 0 close comma  co-vertices ( 0 , ± 8 ) open 0 comma plus minus 8 close
34. foci ( 0 , ± 4 ) , open 0 comma plus minus 4 close comma  co-vertices ( ± 2 , 0 ) open plus minus 2 comma 0 close

35. Miniature Golf The figure below represents a miniature golf green. The green is elliptical with the tee at one focus and the hole at the other.

    

    1. How far is the hole from the tee?
    2. Knowing that the border is elliptical, how should you aim your putt from the tee?

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