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CHAPTER 35
Fig. 35-2
35.11
Describe the graph of the polar equation 0 = 0.
This is simply the line through the polar axis, or the x-axis in rectangular coordinates.
35.12
Write a polar equation for the y-axis.
0 - ir/2 yields the line perpendicular to the polar axis and going through the pole, which is the y-axis.
35.13 Describe the graph of the polar equation r = 2 sin ft
Multiplying both sides by r, we obtain r
2 = 2r sin 0, x
2 + y
2 = 2y, x
2 + y
2 - 2y = 0, x
2 + (y - I)
2 = 1.
Thus, the graph is the circle with center at (0,1) and radius 1.
35.14
Describe the graph of the polar equation r = 4 cos ft
Multiplying both sides by r, we obtain r
2 = 4r cos ft x
2 + y
2 = 4x, x
2 -4x + y
2 = 0, (x - 2)
2 + y
2 = 4.
Thus, the graph is the circle with center at (2,0) and radius 2.
35.15
Describe the graph of the polar equation r = tan 0 sec ft
r = sin 0/cos
2 ft r cos
2 0 = sin ft r
2 cos
2 0 = r sin ft, x
2 = y. Thus, the graph is a parabola.
35.16
Describe the graph of the polar equation r = 6/V9 — 5 sin
2 ft
The graph is an ellipse.
r2 = 36/(9-5sin20), r2(9 - 5 sin2 ft) = 36, 9r2 - 5r2 sin2 0 = 36, 9(x2 + y2) - 5y2 = 36, 9*2 + 4/= 36,
35.17
Describe the graph of the polar equation r = -10 cos ft
Multiply both sides by r: r
2 = -Wr cos 0, x
2 + y
2 = -10*, x
2 + Wx + y
2 = 0, (x + 5)
2 + y
2 = 25.
Thus, the graph is the circle with center (—5,0) and radius 5.
35.18
Describe the graph of the polar equation r = 6(sin 0 + cos 0).
Multiply both sides by r: r
2 = 6r sin 0 + 6r cos ft x
2 + y
2 = 6y + 6x, x
2 -6x + y
2 -6y = 0, (AC - 3)
2 +
(y - 3)
2 = 18. Thus, the graph is the circle with center at (3,3) and radius 3V2.
35.19
Describe the graph of the polar equation r = 5 esc ft
r = 5/sin ft, r sin 0 = 5, y = 5. Thus, the graph is a horizontal line.
35.20
Describe the graph of the polar equation r = — 3 sec ft
r = —3/cos ft r cos 0 = —3, x = —3. Hence, the graph is a vertical line.
35.21
Change the rectangular equation x
2 + y
2 = 16 into a polar equation.
r
2 = 16, r = 4 (or r=-4).
35.22 Change the rectangular equation x
2 — y
2 = 1 into a polar equation.
r
2 cos
2 0-r
2 sin
2 0 = l, r
2
(cos
2 0 -sin
2 0) = 1, r
2 cos20 = 1, r
2 = sec2ft
CHAPTER 35
Fig. 35-2
35.11
Describe the graph of the polar equation 0 = 0.
This is simply the line through the polar axis, or the x-axis in rectangular coordinates.
35.12
Write a polar equation for the y-axis.
0 - ir/2 yields the line perpendicular to the polar axis and going through the pole, which is the y-axis.
35.13 Describe the graph of the polar equation r = 2 sin ft
Multiplying both sides by r, we obtain r
2 = 2r sin 0, x
2 + y
2 = 2y, x
2 + y
2 - 2y = 0, x
2 + (y - I)
2 = 1.
Thus, the graph is the circle with center at (0,1) and radius 1.
35.14
Describe the graph of the polar equation r = 4 cos ft
Multiplying both sides by r, we obtain r
2 = 4r cos ft x
2 + y
2 = 4x, x
2 -4x + y
2 = 0, (x - 2)
2 + y
2 = 4.
Thus, the graph is the circle with center at (2,0) and radius 2.
35.15
Describe the graph of the polar equation r = tan 0 sec ft
r = sin 0/cos
2 ft r cos
2 0 = sin ft r
2 cos
2 0 = r sin ft, x
2 = y. Thus, the graph is a parabola.
35.16
Describe the graph of the polar equation r = 6/V9 — 5 sin
2 ft
The graph is an ellipse.
r2 = 36/(9-5sin20), r2(9 - 5 sin2 ft) = 36, 9r2 - 5r2 sin2 0 = 36, 9(x2 + y2) - 5y2 = 36, 9*2 + 4/= 36,
35.17
Describe the graph of the polar equation r = -10 cos ft
Multiply both sides by r: r
2 = -Wr cos 0, x
2 + y
2 = -10*, x
2 + Wx + y
2 = 0, (x + 5)
2 + y
2 = 25.
Thus, the graph is the circle with center (—5,0) and radius 5.
35.18
Describe the graph of the polar equation r = 6(sin 0 + cos 0).
Multiply both sides by r: r
2 = 6r sin 0 + 6r cos ft x
2 + y
2 = 6y + 6x, x
2 -6x + y
2 -6y = 0, (AC - 3)
2 +
(y - 3)
2 = 18. Thus, the graph is the circle with center at (3,3) and radius 3V2.
35.19
Describe the graph of the polar equation r = 5 esc ft
r = 5/sin ft, r sin 0 = 5, y = 5. Thus, the graph is a horizontal line.
35.20
Describe the graph of the polar equation r = — 3 sec ft
r = —3/cos ft r cos 0 = —3, x = —3. Hence, the graph is a vertical line.
35.21
Change the rectangular equation x
2 + y
2 = 16 into a polar equation.
r
2 = 16, r = 4 (or r=-4).
35.22 Change the rectangular equation x
2 — y
2 = 1 into a polar equation.
r
2 cos
2 0-r
2 sin
2 0 = l, r
2
(cos
2 0 -sin
2 0) = 1, r
2 cos20 = 1, r
2 = sec2ft