James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

appendix c Grpahs of Second-Degree Equations A23
EXAMPLE 8 Sketch the curve x − 1 2 y 2 .
SOLUTIOn This time we start with the parabola x − 2y 2 (as in Figure 6 with a − 21)
and shift one unit to the right to get the graph of x − 1 2 y 2 . (See Figure 17.)
y
y
0
x
0
1
x
FIGURE 17
(a) x=_¥
(b) x=1-¥
■
Exercises
1–4 Find an equation of a circle that satisfies the given
15. 16x 2 2 25y 2 − 400 16. 25x 2 1 4y 2 − 100
plot points. Just use the standard graphs given in Figures 5, 6, 8, 37–40 Sketch the graph of the set.
10, and 11 and shift if necessary.
37. hsx, yd |
11. y − 2x 2 12. y 2 2 x 2 − 1
1 y 2 < 1j
38. hsx, yd | x 2 1 y 2 . 4j
39. hsx, yd | y > x 2 2 1j
40. hsx, yd | x 2 1 4y 2 < 4j
13. x 2 1 4y 2 − 16 14. x − 22y 2
conditions.
1. Center s3, 21d, radius 5
17. 4x 2 1 y 2 − 1 18. y − x 2 1 2
2. Center s22, 28d, radius 10
19. x − y 2 2 1 20. 9x 2 2 25y 2 − 225
3. Center at the origin, passes through s4, 7d
21. 9y 2 2 x 2 − 9 22. 2x 2 1 5y 2 − 10
4. Center s21, 5d, passes through s24, 26d
23. xy − 4
24. y − x 2 1 2x
5–9 Show that the equation represents a circle and find the center
and radius.
25. 9sx 2 1d 2 1 4sy 2 2d 2 − 36
26. 16x 2 1 9y 2 2 36y − 108
27. y − x 2 2 6x 1 13 28. x 2 2 y 2 2 4x 1 3 − 0
5. x 2 1 y 2 2 4x 1 10y 1 13 − 0
29. x − 4 2 y 2 30. y 2 2 2x 1 6y 1 5 − 0
6. x 2 1 y 2 1 6y 1 2 − 0
31. x 2 1 4y 2 2 6x 1 5 − 0
7. x 2 1 y 2 1 x − 0
32. 4x 2 1 9y 2 2 16x 1 54y 1 61 − 0
8. 16x 2 1 16y 2 1 8x 1 32y 1 1 − 0
33–34 Sketch the region bounded by the curves.
9. 2x 2 1 2y 2 2 x 1 y − 1
33. y − 3x, y − x 2 34. y − 4 2 x 2 , x 2 2y − 2
10. Under what condition on the coefficients a, b, and c does the
equation x 2 1 y 2 1 ax 1 by 1 c − 0 represent a circle?
When that condition is satisfied, find the center and radius of
the circle.
11–32 Identify the type of curve and sketch the graph. Do not
35. Find an equation of the parabola with vertex s1, 21d that
passes through the points s21, 3d and s3, 3d.
36. Find an equation of the ellipse with center at the origin that
passes through the points s1, 210s2 y3d and s22, 5s5 y3d.
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