7-4_Rotations_of_Conic_Sections.pdf
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Write each equation in the x′ y ′−plane for the given value of . Then identify the conic. 1. x 2 – y 2 = 9, = SOLUTION: x 2 – y 2 = 9, = Find the equations for x and y . x = x′ cos − y ′ sin x = x′− y ′ y = x′ sin + y ′ cos y = x′+ y ′ Substitute into the original equation. B 2 − 4 AC = 16 which is greater than 0, so the conic is a hyperbola. − = 9 (x') 2 − x'y' − x'y' + ( y' ) 2 − = 9 (x') 2 − x'y' + ( y' ) 2 − (x') 2 − x'y' − ( y' ) 2 = 9 (x') 2 − x'y' + ( y' ) = 9 –(x') 2 − 2 x'y' + ( y' ) = 18 (x′) 2 + 2 x′ y ′– ( y ′) 2 + 18 = 0 ANSWER: (x′) 2 + 2 x′ y ′– ( y ′) 2 + 18 = 0; hyperbola eSolutions Manual - Powered by Cognero Page 1 7-4 Rotations of Conic Sections
Transcript of 7-4_Rotations_of_Conic_Sections.pdf
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Write each equation in the xy plane for the given value of . Then identify the conic.
1.x2 y2 = 9, =
SOLUTION:
x2 y2 = 9, =
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 16 which is greater than 0, so the conic is a hyperbola.
= 9
(x')2 x'y' x'y' + (y')2
= 9
(x')2 x'y' + (y')2 (x')2 x'y' (y')2
= 9
(x')2 x'y' + (y')
= 9
(x')2 2 x'y' + (y') = 18
(x)2 + 2 xy (y )2 + 18 = 0
ANSWER:
(x)2 + 2 xy (y )
2 + 18 = 0; hyperbola
2.xy = 8, =45
SOLUTION:
xy = 8, =45 Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 4 which is greater than 0, so the conic is a hyperbola.
= 8
(x')2 x'y' + x'y' (y')
2 = 8
(x')2 (y')
2 + 8
= 0
(x)2 (y )2 + 16 = 0
ANSWER:
(x)2 (y )2 + 16 = 0; hyperbola
3.x2 8y = 0, =
SOLUTION:
x2 8y = 0, =
Find the equations for x and y .
x = xcos y sin x = y y = xsin +y cos y = x Substitute into the original equation.
There is only one squared term, so the conic is a parabola.
(y')2 8x' = 0
(y')2 8x' = 0
ANSWER:
(y )2 8x = 0; parabola
4.2x2 + 2y2 = 8, =
SOLUTION:
2x2 + 2y
2 = 8, =
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 4 which is less than 0, B = 0, and A = C, so the conic is a circle.
2 + 2
= 8
2 + 2
= 8
(x')2 x'y' + (y)
2 + (x)
2 + x'y' + (y')
2 = 8
2(x)2 + 2(y )2 = 8
(x)2 + (y )
2 4 = 0
ANSWER:
(x)2 + (y )
2 4 = 0; circle
5.y2 + 8x = 0, =30
SOLUTION:
y2 + 8x = 0, =30
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 0, so the conic is a parabola.
+ 8
= 0
(x')2 + x'y' + x'y' + (y')
2 + 4 x' 4y'
= 0
(x')2 + x'y' + (y')
2 + 4 x' 4y'
= 0
(x)2 + 2 xy + 3(y )
2 + 16 x 16y' = 0
ANSWER:
(x)2 + 2 xy + 3(y )2 + 16 x 16y = 0; parabola
6.4x2 + 9y2 = 36, =30
SOLUTION:
4x2 + 9y
2 = 36, =30
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 2304 which is less than 0 and B0,sotheconicisanellipse.
4 + 9
= 36
4 + 9
= 36
3(x')2 2 x'y' + (y')2 + (x')2 + x'y' + (y')2
= 36
12(x')2 8 x'y' + 4(y')2 + 9(x')2 + 18 x'y' + 27(y')2 = 144
21(x)2 + 10 xy + 31(y )2 144 = 0
ANSWER:
21(x)2 + 10 xy + 31(y )
2 144 = 0; ellipse
7.x2 5x + y2 = 3, =45
SOLUTION:
x2 5x + y
2 = 3, =45
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 16 which is less than 0 and B0,sotheconicisanellipse.
5 +
= 3
(x')2 x'y' x'y' + (y')
2 x' + y' + (x')
2 + x'y' + x'y'
+ (y')2
= 3
(x')2 + (y')
2 x' + y' 3
= 0
2(x)2 + 2(y )
2 5 x + 5 y 6 = 0
ANSWER:
2(x)2 + 2(y )2 5 x + 5 y 6 = 0; ellipse
8.49x2 16y2 = 784, =
SOLUTION:
49x2 16y2 = 784, =
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 15,811 which is greater than 0, so the conic is a hyperbola.
49 16
= 784
49 16
= 784
(x')2 49x'y' + (y')2 8(x')2 16x'y' 8(y')2
= 784
49(x')2 98x'y' + 49(y')
2 16(x')
2 32x'y' 16(y')
2 = 1568
33(x)2 130xy + 33(y )2 1568 = 0
ANSWER:
33(x)2 130xy + 33(y )2 1568 = 0; hyperbola
9.4x2 25y2 = 64, =90
SOLUTION:
4x2 25y2 = 64, =90
Find the equations for x and y .
x = xcos y sin x = y y = xsin +y cos y = x Substitute into the original equation.
B2 4AC = 400 which is greater than 0, so the conic is a hyperbola.
4(y')2 25(x')
2 = 64
4(y')2 25(x')2 = 64
25(x)2 4(y )
2 + 64 = 0
ANSWER:
25(x)2 4(y )
2 + 64 = 0; hyperbola
10.6x2 + 5y2 = 30, =30
SOLUTION:
6x2 + 5y
2 = 30, =30
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 1920 which is less than 0 and B0,sotheconicisanellipse.
6 + 5
= 30
6 + 5
= 30
(x')2 3 x'y' + (y')
2 + (x')
2 + x'y' + (y')
2
= 30
18(x')2 12 x'y' + 6(y')
2 + 5(x')
2 + 10 x'y' + 15(y')
2 = 120
23(x)2 2 xy + 21(y )
2 120 = 0
ANSWER:
23(x)2 2 xy + 21(y )2 120 = 0; ellipse
Using a suitable angle of rotation for the conic with each given equation, write the equation in standard form.
11.xy = 4
SOLUTION:xy = 4
B2 4AC = 1 which is greater than 0, so the conic is a hyperbola.
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
cot 2 = 0
2 =
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
xy = 4
= 4
(x)2 xy + xy (y)
2 = 4
(x)2 (y)
2 = 4
(x)2 (y)
2 = 1
= 1
ANSWER:
=1
12.x2 xy + y2 = 2
SOLUTION:
x2 xy + y
2 = 2
B2 4AC = 3 which is less than 0 and B0,sotheconicisanellipse.
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
cot 2 = 0
2 =
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
x2 xy + y
2 = 2
+
= 2
(y)2
= 2
(x)2 + (y)
2
= 2
(x)2 + (y)
2 = 1
=1
ANSWER:
13.145x2 + 120xy + 180y2 = 900
SOLUTION:
145x2 + 120xy + 180y
2 = 900
B2 4AC = 90,000 which is less than 0 and B0,sotheconicisanellipse.
72 + 24
2 = 25
2
sin 2 = andcos2 =
Use the halfangle identities to determine sin andcos .
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
cot 2 =
sin =
cos =
=
=
= =
=
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
145x2 + 120xy + 180y
2 = 900
145 + 120 +180 = 900
145 + 120 +
180
= 900
(x)2 xy + (y)2 + (x)2 xy (y)2 + (x)2 + xy +
(y)2
= 900
261(x)2 696xy + 464(y)
2 + 288(x)
2 168xy 288(y)
2 + 576(x)
2 + 864xy + 324(y)
2 = 4500
1125(x)2 + 500(y)2 = 4500
+
= 1
ANSWER:
+ =1
14.16x2 24xy + 9y2 5x 90y + 25 = 0
SOLUTION:
16x2 24xy + 9y2 5x 90y + 25 = 0
B2 4AC = 0, so the conic is a parabola.
72 + 24
2 = 25
2
sin 2 = andcos2 =
Use the halfangle identities to determine sin andcos .
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
cot 2 =
sin =
cos =
=
=
= =
=
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
16x2 24xy + 9y
2 5x 90y + 25 = 0
16 24 + 9
5 90 + 25
= 0
16 24 +
9 3x + 4y 72x 54y + 25
= 0
(x)2 xy + (y)
2 (x)
2 + xy + (y)
2 + (x)
2
+
xy + (y)2 75x 50y + 25
= 0
144(x)2 384xy + 256(y)2 288(x)2 + 168xy + 288(y)2 + 144(x)2 +
216xy + 81(y)2 1875x 1250y + 625 = 0
625(y)2 1875x 1250y + 625 = 0
(y)2 2y + 1 3x = 0
(y 1)2 = 3x
ANSWER:
(y 1)2 = 3x
15.2x2 72xy + 23y2 + 100x 50y = 0
SOLUTION:
2x2 72xy + 23y
2 + 100x 50y = 0
B2 4AC = 5000 which is greater than 0, so the conic is a hyperbola.
72 + 24
2 = 25
2
sin 2 = andcos2 =
Use the halfangle identities to determine sin andcos .
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
cot 2 =
sin =
cos =
=
=
=
=
=
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
2x2 72xy + 23y2 + 100x 50y = 0
2 72 + 23 + 100
50
= 0
2 72
+23 + 80x 60y 30x 40y
= 0
(x)2 xy + (y)2 (x)2 xy + (y)2 + (x)2 + xy
+
(y)2 + 50x 100y
= 0
32(x)2 48xy + 18(y)
2 864(x)
2 504xy + 864 (y)
2 + 207(x)
2 + 552xy +
368(y)2 + 1250x 2500y = 0
625(x)2 + 1250(y)
2 + 1250x 2500y = 0
(x)2 + 2(y)2 + 2x 4y = 0
2(y)2 4y (x)
2 + 2x = 0
2[(y)2 2y + 1] [(x)2 2x + 1] 2+1 = 0
2(y 1)2 (x 1)
2 = 1
= 1
ANSWER:
16.x2 3y2 8x + 30y = 60
SOLUTION:
x2 3y
2 8x + 30y = 60
B2 4AC = 12 which is greater than 0, so the conic is a hyperbola.
The conic has not been rotated. Convert the equation to standard form.
cot 2 =
cot 2 =
tan 2 = or0
sin 2 = 0
2 = 0 = 0
x2 3y
2 8x + 30y = 60
x2 8x 3y2 + 30y 60 = 0
(x2 8x + 16) 3(y
2 10y + 25) 16 + 3(25) 60 = 0
(x 4)2 3(y 5)2 = 1
= 1
ANSWER:
17.8x2 + 12xy + 3y2 + 4 = 6
SOLUTION:
8x2 + 12xy + 3y
2 + 4 = 6
B2 4AC = 4 which is greater than 0, so the conic is a hyperbola.
=13
sin 2 = andcos2 =
Use the halfangle identities to determine sin andcos .
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
cot 2 =
sin =
cos =
= =
=
=
=
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
8x2 + 12xy + 3y
2 + 4 = 6
8 + 12 + 3
= 2
8 + 12
+ 3
= 2
(x)2 xy + (y)2 + (x)2 + xy (y)2 + (x)2 + xy + (y)2
= 2
72 (x)2 96xy + 32(y)
2 + 72 (x)
2 + 60xy 72 (y)
2 + 12(x)
2 + 36xy + 27(y)
2 = 26
156(x)2 13(y)2 = 26
6(x)2
= 1
= 1
ANSWER:
18.73x2 + 72xy + 52y2 + 25x + 50y 75 = 0
SOLUTION:
73x2 + 72xy + 52y
2 + 25x + 50y 75 = 0
B2 4AC = 10,000 which is less than 0 and B0,sotheconicisanellipse.
72 + 24
2 = 25
2
sin 2 = andcos2 =
Use the halfangle identities to determine sin andcos .
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
=
=
sin =
cos =
=
=
=
=
=
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
73x2 + 72xy + 52y
2 + 25x + 50y 75 = 0
73 + 72 + 52 +
25 + 50 75
= 0
73 + 72 +
52 + 20x 15y + 30x + 40y 75
= 0
(x)2 xy + (y)
2 + (x)
2 + xy (y)
2 +
(x)2 +
xy + (y)2 + 50x + 25y 75
= 0
1168(x)2 1752xy + 657(y)
2 + 864(x)
2 + 504xy 864 (y)
2 + 468(x)
2 +
1248xy + 832(y)2 + 1250x + 625y 1875 = 0
2500(x)2 + 625(y)
2 + 1250x + 625y 1875 = 0
4(x)2 + (y)2 + 2x + y 3 = 0
+(y)2 + y 3
= 0
+ (y)2 + y + 3
= 0
+
=
+
= 1
=1
ANSWER:
=1
Write an equation for each conic in the xyplane for the given equation in xy form and the given value of .
19.(x)2 + 3(y )2 = 8, =
SOLUTION:
(x)2 + 3(y )2 = 8, =
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
(x)2 + 3(y )
2 = 8
+ 3
= 8
x2 + xy + y
2 + 3
= 8
x2 + xy + y
2 + y
2 3xy + x
2 = 8
2x2 2xy + 2y2 = 8
x2 xy + y
2 4 = 0
ANSWER:
x2 xy + y2 4 = 0
20. =1, =
SOLUTION:
=1, =
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
= 1
9(x)2 (y)
2 = 225
9
= 225
9 y2 + xy x
2 = 225
x2 + 9xy + y
2 y
2 + xy x
2 = 225
4x2 + 10xy + 4y
2 225 = 0
ANSWER:
4x2 + 10xy + 4y
2 225 = 0
21. =1, =
SOLUTION:
=1, =
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
= 1
4(x)2 (y)2 = 36
4
= 36
4 y2 + xy x2
= 36
x2 + 2 xy + 3y
2 y2 + xy x2
= 36
4x2 + 8 xy + 12y
2 y2 + 2 xy 3x2 = 144
x2 + 10 xy + 11y
2 144 = 0
ANSWER:
x2 + 10 xy + 11y
2 144 = 0
22.(x)2 = 8(y ), =45
SOLUTION:
(x)2 = 8(y ), =45
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
(x)2 = 8(y)
(x)2 8y = 0
8
= 0
x2 + xy + y
2 4 y + 4 x
= 0
x2 + 2xy+y2 + 8 x 8 y = 0
ANSWER:
x2 + 2xy+y2 + 8 x 8 y = 0
23. + =1, =
SOLUTION:
+ =1, =
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
+
= 1
4(x)2 + (y)
2 = 28
4 +
= 28
4 + y2 xy + x
2
= 28
3x2 + 2 xy + y
2 + y
2 xy + x
2
= 28
12x2 + 8 xy + 4y
2 + 3y
2 2 xy + x
2 = 112
13x2 + 6 xy + 7y
2 112 = 0
ANSWER:
13x2 + 6 xy + 7y
2 112 = 0
24.4x = (y )2, =30
SOLUTION:
4x = (y )2, =30
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
4x = (y)2
4(x)2 (y)
2 = 0
4
= 0
2 x + 2y y2 + xy x
2
= 0
4(2 x + 2y y2 + xy x
2)
= (4)0
8 x 8y + 3y2 2 xy + x
2 = 0
x2 2 xy + 3y
2 8 x 8y = 0
ANSWER:
x2 2 xy + 3y
2 8 x 8y = 0
25. =1, =45
SOLUTION:
=1, =45
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
= 1
(x)2 4(y)2 = 64
4
= 64
x2 + xy + y
2 4
= 64
x2 + xy + y
2 2y2+ 4xy 2x2
= 64
x2 + 2xy + y
2 4y
2+ 8xy 4x
2 = 128
3x2 + 10xy 3y2 = 128
3x2 10xy + 3y
2 + 128 = 0
ANSWER:
3x2 10xy + 3y2 + 128 = 0
26.(x)2 = 5(y ), =
SOLUTION:
(x)2 = 5(y ), =
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
(x)2 = 5(y)
2
(x)2 5(y)2 = 0
5
= 0
x2 + xy + y
2 y + x
= 0
x2 + 2 xy + 3y
2 10y + 10 x = 0
x2 + 3y
2 + 2 xy + 10 x 10y = 0
ANSWER:
x2 + 3y
2 + 2 xy + 10 x 10y = 0
27. =1, =30
SOLUTION:
=1, =30
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
= 1
9(x)2 4(y)
2 = 36
9 4
= 36
9 4
= 36
x2 + xy + y
2 3y
2 + 2 xy x
2
= 36
27x2 + 18 xy + 9y
2 12y
2 + 8 xy 4x
2 = 144
23x2 + 26 xy 3y
2 144 = 0
ANSWER:
23x2 + 26 xy 3y
2 144 = 0
28. + =1, =60
SOLUTION:
+ =1, =60
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
+
= 1
4(x)2 + 3(y)2 = 12
4 + 3
= 12
4 + 3
= 12
x2 + 2 xy + 3y
2 + y
2 xy + x2
= 12
4x2 + 8 xy + 12y
2 + 3y
2 6 xy + 9x2 = 48
13x2 + 2 xy + 15y
2 48 = 0
ANSWER:
13x2 + 2 xy + 15y
2 48 = 0
29.ASTRONOMY Suppose 144(x)2 + 64(y )2= 576 models the shape in the xy plane of a reflecting mirror in a telescope. a.Ifthemirrorhasbeenrotated30,determinetheequationofthemirrorinthexyplane. b. Graph the equation.
SOLUTION:
a. Find the equations for x and y for =30.
x= x cos +y sin
x = x + y
y=y cos x sin
y = y x
Substitute these values into the original equation.
b. Graph the equation by solving for y .
84y2 + (40 x)y + (124x
2 576) = 0
Use the quadratic formula.
y =
y =
Graph the conic using your graphing calculator.
144(x)2 + 64(y )2 = 576
144 + 64
= 576
144 + 64
= 576
108x2 + 36y
2 + 72 xy + 48y
2 + 16x
2 32 xy = 576
124x2 + 40 xy + 84y
2 576 = 0
ANSWER:
a. 31x2 10 xy + 21y2 = 144
b.
Graph each equation if it has been rotated at the indicated angle.
30. + =160
SOLUTION:
+ =160
Find the equations for x and y .
x= x cos +y sin
x = x + y
y=y cos x sin
y = y x
Substitute these values into the original equation.
Graph the equation by solving for y .
31y2 + (10 x)y + 21x
2 144
Use the quadratic formula.
y =
y =
y =
Graph the conic using your graphing calculator.
+
= 1
9(x)2 + 4(y)
2 = 36
9 + 4
= 36
9 + 4
= 36
x2 + xy + y
2 + y
2 2 xy + 3x
2
= 36
9x2 + 18 xy + 27y
2 + 4y
2 8 xy + 12x
2 = 144
21x2 + 10 xy + 31y
2 144 = 0
ANSWER:
31. =1; 45
SOLUTION:
=1; 45
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
Graph the equation by solving for y .
7y2 (122x)y + 7x2 + 1800 = 0
Use the quadratic formula.
y =
y =
y =
Graph the conic using your graphing calculator.
= 1
36(x)2 25(y)2 = 900
36 25
= 900
36 25
= 900
9x2 + 36xy + 9y
2 y2 + 25xy x2
= 900
18x2 + 72xy + 18y
2 25y
2 + 50xy 25x
2 = 1800
7x2 + 122xy 7y2 1800 = 0
7x2 122xy + 7y
2 + 1800 = 0
ANSWER:
32.(x)2 + 6x y =9; 30
SOLUTION:
(x)2 + 6x y =9; 30
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.
x = x + y
y=y cos x sin
y = y x
Substitute these values into the original equation.
Graph the equation by solving for y .
y2 + (2 x + 12 2 )y + 3x
2 + (12 + 2)x + 36 = 0
Use the quadratic formula.
y =
y =
y =
Graph the conic using your graphing calculator.
(x)2 + 6x y = 9
+ 6 y + x
= 9
x2 + xy + y
2 + 3 x + 3y y + x
= 9
3x2 + 2 xy + y
2 + 12 x + 12y 2 y + 2x = 36
3x2 + 2 xy + y
2 + (12 + 2)x + (12 2 )y + 36 = 0
ANSWER:
33.8(x)2 + 6(y )2 = 24; 30
SOLUTION:
8(x)2 + 6(y )
2 = 24; 30
x = x + y
y=y cos x sin
y = y x
Substitute these values into the original equation.
Graph the equation by solving for y .
13y2 + (2 x) y + 15x
2 48 = 0
Use the quadratic formula.
y =
y =
y =
8(x)2 + 6(y )2 = 24
8 + 6
= 24
8 + 6
= 24
6x2 + 4 xy + 2y
2 + y
2 3 xy + x2
= 24
12x2 + 8 xy + 4y
2 + 9y
2 6 xy + 3x2 = 48
15x2 + 2 xy + 13y
2 48 = 0
ANSWER:
34. =1; 45
SOLUTION:
=1; 45
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
Graph the equation by solving for y .
3y2 + (10x)y + 3x
2 32 = 0
y =
y =
y =
= 1
4(x)2 (y)
2 = 16
4
= 16
4 y2 + xy x
2 = 16
2x2 + 4xy +2y
2 y
2 + xy x
2 = 16
4x2 + 8xy + 4y
2 y2 + 2xy x2 = 32
3x2 + 10xy + 3y
2 32 = 0
ANSWER:
35.y = 3(x)2 2x+560
SOLUTION:
y = 3(x)2 2x+560
Find the equations for x and y .
x= x cos +y sin
x = x + y
y=y cos x sin
y = y x
Substitute these values into the original equation.
Graph the equation by solving for y .
9y2 + (2 + 4 6 x)y + (3x2 + (4 2 )x 20) = 0
y =
y =
y =
y = 3(x)2 2x+5
y 3(x)2 + 2x 5 = 0
y x 3 + 2 5
= 0
y x 3 + x + y 5
= 0
y x x2 xy y2 + x + y 5
= 0
2y 2 x 3x2 6 xy 9y2 + 4x + 4 y 20 = 0
3x2 6 xy 9y2 + (4 2 )x + (2 + 4 y 20 = 0
ANSWER:
36.COMMUNICATIONAsatellitedishtracksasatellitedirectlyoverhead.Supposey = x2 models the shape of
the dish when it is oriented in this position. Later in the day, the dish is observed to have rotated approximately 30. a. Write an equation that models the new orientation of the dish. b. Use a graphing calculator to graph both equations on the same screen. Sketch this graph on your paper.
SOLUTION:
a. y = x2, =30
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
b.
y = x2
x+ y
=
x+ y
=
x+ y
= (x)2 xy + (y)2
12x + 12 y = 3(x)2 2 xy + (y)2
0 = 3(x)2 2 xy + (y )2 12x 12 y
ANSWER:
a. 3(x)2 2 xy + (y )2 12x 12 y = 0
b.
GRAPHING CALCULATOR Graph the conic given by each equation.
37.x2 2xy + y2 5x 5y = 0
SOLUTION:Graph the equation by solving for y .
y =
y =
y =
x2 2xy + y2 5x 5y = 0
y2 + (2x 5)y + x
2 5x = 0
ANSWER:
38.2x2 + 9xy + 14y2 = 5
SOLUTION:Graph the equation by solving for y .
y =
y =
2x2 + 9xy + 14y
2 = 5
14y2 + (9x)y + 2x 5 = 0
ANSWER:
39.8x2 + 5xy 4y2 = 2
SOLUTION:Graph the equation by solving for y .
8x2 + 5xy 4y2 = 2
4y2 + (5x)y + 8x
2 + 2 = 0
y =
y =
y =
ANSWER:
40.2x2 + 4 xy + 6y2 + 3x = y
SOLUTION:Graph the equation by solving for y .
2x2 + 4 xy + 6y
2 + 3x = y
6y2 + (4 x 1)y + 2x
2 + 3x = 0
y =
y =
y =
ANSWER:
41.2x2 + 4xy + 2y2 + 2 x 2 y = 12
SOLUTION:Graph the equation by solving for y .
y =
y =
y =
2x2 + 4xy + 2y
2 + 2 x 2 y = 12
2y2 + (4x 2 )y + 2x
2 + 2 x + 12 = 0
ANSWER:
42.9x2 + 4xy + 6y2 = 20
SOLUTION:Graph the equation by solving for y .
y =
y =
y =
9x2 + 4xy + 6y
2 = 20
6y2 + (4x)y + 9x
2 20 = 0
ANSWER:
43.x2 + 10 xy + 11y2 64 = 0
SOLUTION:Graph the equation by solving for y .
y =
y =
x2 + 10 xy + 11y
2 64 = 0
11y2 + (10 x)y + x
2 64 = 0
ANSWER:
44.x2 + y2 4 = 0
SOLUTION:Graph the equation by solving for y .
y =
y =
y =
x2 + y
2 4 = 0
y2 + x
2 4 = 0
ANSWER:
45.x2 2 xy y2 + 18 = 0
SOLUTION:Graph the equation by solving for y .
x2 2 xy y
2 + 18 = 0
y2 (2 x)y + x
2 + 18 = 0
y =
y =
y = x
ANSWER:
46.2x2 + 9xy + 14y2 5 = 0
SOLUTION:Graph the equation by solving for y .
y =
y =
2x2 + 9xy + 14y
2 5 = 0
14y2 + (9x)y + 2x
2 5 = 0
eSolutions Manual - Powered by Cognero Page 1
7-4 Rotations of Conic Sections
-
Write each equation in the xy plane for the given value of . Then identify the conic.
1.x2 y2 = 9, =
SOLUTION:
x2 y2 = 9, =
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 16 which is greater than 0, so the conic is a hyperbola.
= 9
(x')2 x'y' x'y' + (y')2
= 9
(x')2 x'y' + (y')2 (x')2 x'y' (y')2
= 9
(x')2 x'y' + (y')
= 9
(x')2 2 x'y' + (y') = 18
(x)2 + 2 xy (y )2 + 18 = 0
ANSWER:
(x)2 + 2 xy (y )
2 + 18 = 0; hyperbola
2.xy = 8, =45
SOLUTION:
xy = 8, =45 Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 4 which is greater than 0, so the conic is a hyperbola.
= 8
(x')2 x'y' + x'y' (y')
2 = 8
(x')2 (y')
2 + 8
= 0
(x)2 (y )2 + 16 = 0
ANSWER:
(x)2 (y )2 + 16 = 0; hyperbola
3.x2 8y = 0, =
SOLUTION:
x2 8y = 0, =
Find the equations for x and y .
x = xcos y sin x = y y = xsin +y cos y = x Substitute into the original equation.
There is only one squared term, so the conic is a parabola.
(y')2 8x' = 0
(y')2 8x' = 0
ANSWER:
(y )2 8x = 0; parabola
4.2x2 + 2y2 = 8, =
SOLUTION:
2x2 + 2y
2 = 8, =
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 4 which is less than 0, B = 0, and A = C, so the conic is a circle.
2 + 2
= 8
2 + 2
= 8
(x')2 x'y' + (y)
2 + (x)
2 + x'y' + (y')
2 = 8
2(x)2 + 2(y )2 = 8
(x)2 + (y )
2 4 = 0
ANSWER:
(x)2 + (y )
2 4 = 0; circle
5.y2 + 8x = 0, =30
SOLUTION:
y2 + 8x = 0, =30
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 0, so the conic is a parabola.
+ 8
= 0
(x')2 + x'y' + x'y' + (y')
2 + 4 x' 4y'
= 0
(x')2 + x'y' + (y')
2 + 4 x' 4y'
= 0
(x)2 + 2 xy + 3(y )
2 + 16 x 16y' = 0
ANSWER:
(x)2 + 2 xy + 3(y )2 + 16 x 16y = 0; parabola
6.4x2 + 9y2 = 36, =30
SOLUTION:
4x2 + 9y
2 = 36, =30
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 2304 which is less than 0 and B0,sotheconicisanellipse.
4 + 9
= 36
4 + 9
= 36
3(x')2 2 x'y' + (y')2 + (x')2 + x'y' + (y')2
= 36
12(x')2 8 x'y' + 4(y')2 + 9(x')2 + 18 x'y' + 27(y')2 = 144
21(x)2 + 10 xy + 31(y )2 144 = 0
ANSWER:
21(x)2 + 10 xy + 31(y )
2 144 = 0; ellipse
7.x2 5x + y2 = 3, =45
SOLUTION:
x2 5x + y
2 = 3, =45
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 16 which is less than 0 and B0,sotheconicisanellipse.
5 +
= 3
(x')2 x'y' x'y' + (y')
2 x' + y' + (x')
2 + x'y' + x'y'
+ (y')2
= 3
(x')2 + (y')
2 x' + y' 3
= 0
2(x)2 + 2(y )
2 5 x + 5 y 6 = 0
ANSWER:
2(x)2 + 2(y )2 5 x + 5 y 6 = 0; ellipse
8.49x2 16y2 = 784, =
SOLUTION:
49x2 16y2 = 784, =
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 15,811 which is greater than 0, so the conic is a hyperbola.
49 16
= 784
49 16
= 784
(x')2 49x'y' + (y')2 8(x')2 16x'y' 8(y')2
= 784
49(x')2 98x'y' + 49(y')
2 16(x')
2 32x'y' 16(y')
2 = 1568
33(x)2 130xy + 33(y )2 1568 = 0
ANSWER:
33(x)2 130xy + 33(y )2 1568 = 0; hyperbola
9.4x2 25y2 = 64, =90
SOLUTION:
4x2 25y2 = 64, =90
Find the equations for x and y .
x = xcos y sin x = y y = xsin +y cos y = x Substitute into the original equation.
B2 4AC = 400 which is greater than 0, so the conic is a hyperbola.
4(y')2 25(x')
2 = 64
4(y')2 25(x')2 = 64
25(x)2 4(y )
2 + 64 = 0
ANSWER:
25(x)2 4(y )
2 + 64 = 0; hyperbola
10.6x2 + 5y2 = 30, =30
SOLUTION:
6x2 + 5y
2 = 30, =30
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 1920 which is less than 0 and B0,sotheconicisanellipse.
6 + 5
= 30
6 + 5
= 30
(x')2 3 x'y' + (y')
2 + (x')
2 + x'y' + (y')
2
= 30
18(x')2 12 x'y' + 6(y')
2 + 5(x')
2 + 10 x'y' + 15(y')
2 = 120
23(x)2 2 xy + 21(y )
2 120 = 0
ANSWER:
23(x)2 2 xy + 21(y )2 120 = 0; ellipse
Using a suitable angle of rotation for the conic with each given equation, write the equation in standard form.
11.xy = 4
SOLUTION:xy = 4
B2 4AC = 1 which is greater than 0, so the conic is a hyperbola.
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
cot 2 = 0
2 =
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
xy = 4
= 4
(x)2 xy + xy (y)
2 = 4
(x)2 (y)
2 = 4
(x)2 (y)
2 = 1
= 1
ANSWER:
=1
12.x2 xy + y2 = 2
SOLUTION:
x2 xy + y
2 = 2
B2 4AC = 3 which is less than 0 and B0,sotheconicisanellipse.
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
cot 2 = 0
2 =
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
x2 xy + y
2 = 2
+
= 2
(y)2
= 2
(x)2 + (y)
2
= 2
(x)2 + (y)
2 = 1
=1
ANSWER:
13.145x2 + 120xy + 180y2 = 900
SOLUTION:
145x2 + 120xy + 180y
2 = 900
B2 4AC = 90,000 which is less than 0 and B0,sotheconicisanellipse.
72 + 24
2 = 25
2
sin 2 = andcos2 =
Use the halfangle identities to determine sin andcos .
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
cot 2 =
sin =
cos =
=
=
= =
=
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
145x2 + 120xy + 180y
2 = 900
145 + 120 +180 = 900
145 + 120 +
180
= 900
(x)2 xy + (y)2 + (x)2 xy (y)2 + (x)2 + xy +
(y)2
= 900
261(x)2 696xy + 464(y)
2 + 288(x)
2 168xy 288(y)
2 + 576(x)
2 + 864xy + 324(y)
2 = 4500
1125(x)2 + 500(y)2 = 4500
+
= 1
ANSWER:
+ =1
14.16x2 24xy + 9y2 5x 90y + 25 = 0
SOLUTION:
16x2 24xy + 9y2 5x 90y + 25 = 0
B2 4AC = 0, so the conic is a parabola.
72 + 24
2 = 25
2
sin 2 = andcos2 =
Use the halfangle identities to determine sin andcos .
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
cot 2 =
sin =
cos =
=
=
= =
=
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
16x2 24xy + 9y
2 5x 90y + 25 = 0
16 24 + 9
5 90 + 25
= 0
16 24 +
9 3x + 4y 72x 54y + 25
= 0
(x)2 xy + (y)
2 (x)
2 + xy + (y)
2 + (x)
2
+
xy + (y)2 75x 50y + 25
= 0
144(x)2 384xy + 256(y)2 288(x)2 + 168xy + 288(y)2 + 144(x)2 +
216xy + 81(y)2 1875x 1250y + 625 = 0
625(y)2 1875x 1250y + 625 = 0
(y)2 2y + 1 3x = 0
(y 1)2 = 3x
ANSWER:
(y 1)2 = 3x
15.2x2 72xy + 23y2 + 100x 50y = 0
SOLUTION:
2x2 72xy + 23y
2 + 100x 50y = 0
B2 4AC = 5000 which is greater than 0, so the conic is a hyperbola.
72 + 24
2 = 25
2
sin 2 = andcos2 =
Use the halfangle identities to determine sin andcos .
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
cot 2 =
sin =
cos =
=
=
=
=
=
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
2x2 72xy + 23y2 + 100x 50y = 0
2 72 + 23 + 100
50
= 0
2 72
+23 + 80x 60y 30x 40y
= 0
(x)2 xy + (y)2 (x)2 xy + (y)2 + (x)2 + xy
+
(y)2 + 50x 100y
= 0
32(x)2 48xy + 18(y)
2 864(x)
2 504xy + 864 (y)
2 + 207(x)
2 + 552xy +
368(y)2 + 1250x 2500y = 0
625(x)2 + 1250(y)
2 + 1250x 2500y = 0
(x)2 + 2(y)2 + 2x 4y = 0
2(y)2 4y (x)
2 + 2x = 0
2[(y)2 2y + 1] [(x)2 2x + 1] 2+1 = 0
2(y 1)2 (x 1)
2 = 1
= 1
ANSWER:
16.x2 3y2 8x + 30y = 60
SOLUTION:
x2 3y
2 8x + 30y = 60
B2 4AC = 12 which is greater than 0, so the conic is a hyperbola.
The conic has not been rotated. Convert the equation to standard form.
cot 2 =
cot 2 =
tan 2 = or0
sin 2 = 0
2 = 0 = 0
x2 3y
2 8x + 30y = 60
x2 8x 3y2 + 30y 60 = 0
(x2 8x + 16) 3(y
2 10y + 25) 16 + 3(25) 60 = 0
(x 4)2 3(y 5)2 = 1
= 1
ANSWER:
17.8x2 + 12xy + 3y2 + 4 = 6
SOLUTION:
8x2 + 12xy + 3y
2 + 4 = 6
B2 4AC = 4 which is greater than 0, so the conic is a hyperbola.
=13
sin 2 = andcos2 =
Use the halfangle identities to determine sin andcos .
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
cot 2 =
sin =
cos =
= =
=
=
=
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
8x2 + 12xy + 3y
2 + 4 = 6
8 + 12 + 3
= 2
8 + 12
+ 3
= 2
(x)2 xy + (y)2 + (x)2 + xy (y)2 + (x)2 + xy + (y)2
= 2
72 (x)2 96xy + 32(y)
2 + 72 (x)
2 + 60xy 72 (y)
2 + 12(x)
2 + 36xy + 27(y)
2 = 26
156(x)2 13(y)2 = 26
6(x)2
= 1
= 1
ANSWER:
18.73x2 + 72xy + 52y2 + 25x + 50y 75 = 0
SOLUTION:
73x2 + 72xy + 52y
2 + 25x + 50y 75 = 0
B2 4AC = 10,000 which is less than 0 and B0,sotheconicisanellipse.
72 + 24
2 = 25
2
sin 2 = andcos2 =
Use the halfangle identities to determine sin andcos .
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
=
=
sin =
cos =
=
=
=
=
=
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
73x2 + 72xy + 52y
2 + 25x + 50y 75 = 0
73 + 72 + 52 +
25 + 50 75
= 0
73 + 72 +
52 + 20x 15y + 30x + 40y 75
= 0
(x)2 xy + (y)
2 + (x)
2 + xy (y)
2 +
(x)2 +
xy + (y)2 + 50x + 25y 75
= 0
1168(x)2 1752xy + 657(y)
2 + 864(x)
2 + 504xy 864 (y)
2 + 468(x)
2 +
1248xy + 832(y)2 + 1250x + 625y 1875 = 0
2500(x)2 + 625(y)
2 + 1250x + 625y 1875 = 0
4(x)2 + (y)2 + 2x + y 3 = 0
+(y)2 + y 3
= 0
+ (y)2 + y + 3
= 0
+
=
+
= 1
=1
ANSWER:
=1
Write an equation for each conic in the xyplane for the given equation in xy form and the given value of .
19.(x)2 + 3(y )2 = 8, =
SOLUTION:
(x)2 + 3(y )2 = 8, =
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
(x)2 + 3(y )
2 = 8
+ 3
= 8
x2 + xy + y
2 + 3
= 8
x2 + xy + y
2 + y
2 3xy + x
2 = 8
2x2 2xy + 2y2 = 8
x2 xy + y
2 4 = 0
ANSWER:
x2 xy + y2 4 = 0
20. =1, =
SOLUTION:
=1, =
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
= 1
9(x)2 (y)
2 = 225
9
= 225
9 y2 + xy x
2 = 225
x2 + 9xy + y
2 y
2 + xy x
2 = 225
4x2 + 10xy + 4y
2 225 = 0
ANSWER:
4x2 + 10xy + 4y
2 225 = 0
21. =1, =
SOLUTION:
=1, =
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
= 1
4(x)2 (y)2 = 36
4
= 36
4 y2 + xy x2
= 36
x2 + 2 xy + 3y
2 y2 + xy x2
= 36
4x2 + 8 xy + 12y
2 y2 + 2 xy 3x2 = 144
x2 + 10 xy + 11y
2 144 = 0
ANSWER:
x2 + 10 xy + 11y
2 144 = 0
22.(x)2 = 8(y ), =45
SOLUTION:
(x)2 = 8(y ), =45
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
(x)2 = 8(y)
(x)2 8y = 0
8
= 0
x2 + xy + y
2 4 y + 4 x
= 0
x2 + 2xy+y2 + 8 x 8 y = 0
ANSWER:
x2 + 2xy+y2 + 8 x 8 y = 0
23. + =1, =
SOLUTION:
+ =1, =
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
+
= 1
4(x)2 + (y)
2 = 28
4 +
= 28
4 + y2 xy + x
2
= 28
3x2 + 2 xy + y
2 + y
2 xy + x
2
= 28
12x2 + 8 xy + 4y
2 + 3y
2 2 xy + x
2 = 112
13x2 + 6 xy + 7y
2 112 = 0
ANSWER:
13x2 + 6 xy + 7y
2 112 = 0
24.4x = (y )2, =30
SOLUTION:
4x = (y )2, =30
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
4x = (y)2
4(x)2 (y)
2 = 0
4
= 0
2 x + 2y y2 + xy x
2
= 0
4(2 x + 2y y2 + xy x
2)
= (4)0
8 x 8y + 3y2 2 xy + x
2 = 0
x2 2 xy + 3y
2 8 x 8y = 0
ANSWER:
x2 2 xy + 3y
2 8 x 8y = 0
25. =1, =45
SOLUTION:
=1, =45
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
= 1
(x)2 4(y)2 = 64
4
= 64
x2 + xy + y
2 4
= 64
x2 + xy + y
2 2y2+ 4xy 2x2
= 64
x2 + 2xy + y
2 4y
2+ 8xy 4x
2 = 128
3x2 + 10xy 3y2 = 128
3x2 10xy + 3y
2 + 128 = 0
ANSWER:
3x2 10xy + 3y2 + 128 = 0
26.(x)2 = 5(y ), =
SOLUTION:
(x)2 = 5(y ), =
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
(x)2 = 5(y)
2
(x)2 5(y)2 = 0
5
= 0
x2 + xy + y
2 y + x
= 0
x2 + 2 xy + 3y
2 10y + 10 x = 0
x2 + 3y
2 + 2 xy + 10 x 10y = 0
ANSWER:
x2 + 3y
2 + 2 xy + 10 x 10y = 0
27. =1, =30
SOLUTION:
=1, =30
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
= 1
9(x)2 4(y)
2 = 36
9 4
= 36
9 4
= 36
x2 + xy + y
2 3y
2 + 2 xy x
2
= 36
27x2 + 18 xy + 9y
2 12y
2 + 8 xy 4x
2 = 144
23x2 + 26 xy 3y
2 144 = 0
ANSWER:
23x2 + 26 xy 3y
2 144 = 0
28. + =1, =60
SOLUTION:
+ =1, =60
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
+
= 1
4(x)2 + 3(y)2 = 12
4 + 3
= 12
4 + 3
= 12
x2 + 2 xy + 3y
2 + y
2 xy + x2
= 12
4x2 + 8 xy + 12y
2 + 3y
2 6 xy + 9x2 = 48
13x2 + 2 xy + 15y
2 48 = 0
ANSWER:
13x2 + 2 xy + 15y
2 48 = 0
29.ASTRONOMY Suppose 144(x)2 + 64(y )2= 576 models the shape in the xy plane of a reflecting mirror in a telescope. a.Ifthemirrorhasbeenrotated30,determinetheequationofthemirrorinthexyplane. b. Graph the equation.
SOLUTION:
a. Find the equations for x and y for =30.
x= x cos +y sin
x = x + y
y=y cos x sin
y = y x
Substitute these values into the original equation.
b. Graph the equation by solving for y .
84y2 + (40 x)y + (124x
2 576) = 0
Use the quadratic formula.
y =
y =
Graph the conic using your graphing calculator.
144(x)2 + 64(y )2 = 576
144 + 64
= 576
144 + 64
= 576
108x2 + 36y
2 + 72 xy + 48y
2 + 16x
2 32 xy = 576
124x2 + 40 xy + 84y
2 576 = 0
ANSWER:
a. 31x2 10 xy + 21y2 = 144
b.
Graph each equation if it has been rotated at the indicated angle.
30. + =160
SOLUTION:
+ =160
Find the equations for x and y .
x= x cos +y sin
x = x + y
y=y cos x sin
y = y x
Substitute these values into the original equation.
Graph the equation by solving for y .
31y2 + (10 x)y + 21x
2 144
Use the quadratic formula.
y =
y =
y =
Graph the conic using your graphing calculator.
+
= 1
9(x)2 + 4(y)
2 = 36
9 + 4
= 36
9 + 4
= 36
x2 + xy + y
2 + y
2 2 xy + 3x
2
= 36
9x2 + 18 xy + 27y
2 + 4y
2 8 xy + 12x
2 = 144
21x2 + 10 xy + 31y
2 144 = 0
ANSWER:
31. =1; 45
SOLUTION:
=1; 45
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
Graph the equation by solving for y .
7y2 (122x)y + 7x2 + 1800 = 0
Use the quadratic formula.
y =
y =
y =
Graph the conic using your graphing calculator.
= 1
36(x)2 25(y)2 = 900
36 25
= 900
36 25
= 900
9x2 + 36xy + 9y
2 y2 + 25xy x2
= 900
18x2 + 72xy + 18y
2 25y
2 + 50xy 25x
2 = 1800
7x2 + 122xy 7y2 1800 = 0
7x2 122xy + 7y
2 + 1800 = 0
ANSWER:
32.(x)2 + 6x y =9; 30
SOLUTION:
(x)2 + 6x y =9; 30
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.
x = x + y
y=y cos x sin
y = y x
Substitute these values into the original equation.
Graph the equation by solving for y .
y2 + (2 x + 12 2 )y + 3x
2 + (12 + 2)x + 36 = 0
Use the quadratic formula.
y =
y =
y =
Graph the conic using your graphing calculator.
(x)2 + 6x y = 9
+ 6 y + x
= 9
x2 + xy + y
2 + 3 x + 3y y + x
= 9
3x2 + 2 xy + y
2 + 12 x + 12y 2 y + 2x = 36
3x2 + 2 xy + y
2 + (12 + 2)x + (12 2 )y + 36 = 0
ANSWER:
33.8(x)2 + 6(y )2 = 24; 30
SOLUTION:
8(x)2 + 6(y )
2 = 24; 30
x = x + y
y=y cos x sin
y = y x
Substitute these values into the original equation.
Graph the equation by solving for y .
13y2 + (2 x) y + 15x
2 48 = 0
Use the quadratic formula.
y =
y =
y =
8(x)2 + 6(y )2 = 24
8 + 6
= 24
8 + 6
= 24
6x2 + 4 xy + 2y
2 + y
2 3 xy + x2
= 24
12x2 + 8 xy + 4y
2 + 9y
2 6 xy + 3x2 = 48
15x2 + 2 xy + 13y
2 48 = 0
ANSWER:
34. =1; 45
SOLUTION:
=1; 45
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
Graph the equation by solving for y .
3y2 + (10x)y + 3x
2 32 = 0
y =
y =
y =
= 1
4(x)2 (y)
2 = 16
4
= 16
4 y2 + xy x
2 = 16
2x2 + 4xy +2y
2 y
2 + xy x
2 = 16
4x2 + 8xy + 4y
2 y2 + 2xy x2 = 32
3x2 + 10xy + 3y
2 32 = 0
ANSWER:
35.y = 3(x)2 2x+560
SOLUTION:
y = 3(x)2 2x+560
Find the equations for x and y .
x= x cos +y sin
x = x + y
y=y cos x sin
y = y x
Substitute these values into the original equation.
Graph the equation by solving for y .
9y2 + (2 + 4 6 x)y + (3x2 + (4 2 )x 20) = 0
y =
y =
y =
y = 3(x)2 2x+5
y 3(x)2 + 2x 5 = 0
y x 3 + 2 5
= 0
y x 3 + x + y 5
= 0
y x x2 xy y2 + x + y 5
= 0
2y 2 x 3x2 6 xy 9y2 + 4x + 4 y 20 = 0
3x2 6 xy 9y2 + (4 2 )x + (2 + 4 y 20 = 0
ANSWER:
36.COMMUNICATIONAsatellitedishtracksasatellitedirectlyoverhead.Supposey = x2 models the shape of
the dish when it is oriented in this position. Later in the day, the dish is observed to have rotated approximately 30. a. Write an equation that models the new orientation of the dish. b. Use a graphing calculator to graph both equations on the same screen. Sketch this graph on your paper.
SOLUTION:
a. y = x2, =30
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
b.
y = x2
x+ y
=
x+ y
=
x+ y
= (x)2 xy + (y)2
12x + 12 y = 3(x)2 2 xy + (y)2
0 = 3(x)2 2 xy + (y )2 12x 12 y
ANSWER:
a. 3(x)2 2 xy + (y )2 12x 12 y = 0
b.
GRAPHING CALCULATOR Graph the conic given by each equation.
37.x2 2xy + y2 5x 5y = 0
SOLUTION:Graph the equation by solving for y .
y =
y =
y =
x2 2xy + y2 5x 5y = 0
y2 + (2x 5)y + x
2 5x = 0
ANSWER:
38.2x2 + 9xy + 14y2 = 5
SOLUTION:Graph the equation by solving for y .
y =
y =
2x2 + 9xy + 14y
2 = 5
14y2 + (9x)y + 2x 5 = 0
ANSWER:
39.8x2 + 5xy 4y2 = 2
SOLUTION:Graph the equation by solving for y .
8x2 + 5xy 4y2 = 2
4y2 + (5x)y + 8x
2 + 2 = 0
y =
y =
y =
ANSWER:
40.2x2 + 4 xy + 6y2 + 3x = y
SOLUTION:Graph the equation by solving for y .
2x2 + 4 xy + 6y
2 + 3x = y
6y2 + (4 x 1)y + 2x
2 + 3x = 0
y =
y =
y =
ANSWER:
41.2x2 + 4xy + 2y2 + 2 x 2 y = 12
SOLUTION:Graph the equation by solving for y .
y =
y =
y =
2x2 + 4xy + 2y
2 + 2 x 2 y = 12
2y2 + (4x 2 )y + 2x
2 + 2 x + 12 = 0
ANSWER:
42.9x2 + 4xy + 6y2 = 20
SOLUTION:Graph the equation by solving for y .
y =
y =
y =
9x2 + 4xy + 6y
2 = 20
6y2 + (4x)y + 9x
2 20 = 0
ANSWER:
43.x2 + 10 xy + 11y2 64 = 0
SOLUTION:Graph the equation by solving for y .
y =
y =
x2 + 10 xy + 11y
2 64 = 0
11y2 + (10 x)y + x
2 64 = 0
ANSWER:
44.x2 + y2 4 = 0
SOLUTION:Graph the equation by solving for y .
y =
y =
y =
x2 + y
2 4 = 0
y2 + x
2 4 = 0
ANSWER:
45.x2 2 xy y2 + 18 = 0
SOLUTION:Graph the equation by solving for y .
x2 2 xy y
2 + 18 = 0
y2 (2 x)y + x
2 + 18 = 0
y =
y =
y = x
ANSWER:
46.2x2 + 9xy + 14y2 5 = 0
SOLUTION:Graph the equation by solving for y .
y =
y =
2x2 + 9xy + 14y
2 5 = 0
14y2 + (9x)y + 2x
2 5 = 0
eSolutions Manual - Powered by Cognero Page 2
7-4 Rotations of Conic Sections
-
Write each equation in the xy plane for the given value of . Then identify the conic.
1.x2 y2 = 9, =
SOLUTION:
x2 y2 = 9, =
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 16 which is greater than 0, so the conic is a hyperbola.
= 9
(x')2 x'y' x'y' + (y')2
= 9
(x')2 x'y' + (y')2 (x')2 x'y' (y')2
= 9
(x')2 x'y' + (y')
= 9
(x')2 2 x'y' + (y') = 18
(x)2 + 2 xy (y )2 + 18 = 0
ANSWER:
(x)2 + 2 xy (y )
2 + 18 = 0; hyperbola
2.xy = 8, =45
SOLUTION:
xy = 8, =45 Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 4 which is greater than 0, so the conic is a hyperbola.
= 8
(x')2 x'y' + x'y' (y')
2 = 8
(x')2 (y')
2 + 8
= 0
(x)2 (y )2 + 16 = 0
ANSWER:
(x)2 (y )2 + 16 = 0; hyperbola
3.x2 8y = 0, =
SOLUTION:
x2 8y = 0, =
Find the equations for x and y .
x = xcos y sin x = y y = xsin +y cos y = x Substitute into the original equation.
There is only one squared term, so the conic is a parabola.
(y')2 8x' = 0
(y')2 8x' = 0
ANSWER:
(y )2 8x = 0; parabola
4.2x2 + 2y2 = 8, =
SOLUTION:
2x2 + 2y
2 = 8, =
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 4 which is less than 0, B = 0, and A = C, so the conic is a circle.
2 + 2
= 8
2 + 2
= 8
(x')2 x'y' + (y)
2 + (x)
2 + x'y' + (y')
2 = 8
2(x)2 + 2(y )2 = 8
(x)2 + (y )
2 4 = 0
ANSWER:
(x)2 + (y )
2 4 = 0; circle
5.y2 + 8x = 0, =30
SOLUTION:
y2 + 8x = 0, =30
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 0, so the conic is a parabola.
+ 8
= 0
(x')2 + x'y' + x'y' + (y')
2 + 4 x' 4y'
= 0
(x')2 + x'y' + (y')
2 + 4 x' 4y'
= 0
(x)2 + 2 xy + 3(y )
2 + 16 x 16y' = 0
ANSWER:
(x)2 + 2 xy + 3(y )2 + 16 x 16y = 0; parabola
6.4x2 + 9y2 = 36, =30
SOLUTION:
4x2 + 9y
2 = 36, =30
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 2304 which is less than 0 and B0,sotheconicisanellipse.
4 + 9
= 36
4 + 9
= 36
3(x')2 2 x'y' + (y')2 + (x')2 + x'y' + (y')2
= 36
12(x')2 8 x'y' + 4(y')2 + 9(x')2 + 18 x'y' + 27(y')2 = 144
21(x)2 + 10 xy + 31(y )2 144 = 0
ANSWER:
21(x)2 + 10 xy + 31(y )
2 144 = 0; ellipse
7.x2 5x + y2 = 3, =45
SOLUTION:
x2 5x + y
2 = 3, =45
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 16 which is less than 0 and B0,sotheconicisanellipse.
5 +
= 3
(x')2 x'y' x'y' + (y')
2 x' + y' + (x')
2 + x'y' + x'y'
+ (y')2
= 3
(x')2 + (y')
2 x' + y' 3
= 0
2(x)2 + 2(y )
2 5 x + 5 y 6 = 0
ANSWER:
2(x)2 + 2(y )2 5 x + 5 y 6 = 0; ellipse
8.49x2 16y2 = 784, =
SOLUTION:
49x2 16y2 = 784, =
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 15,811 which is greater than 0, so the conic is a hyperbola.
49 16
= 784
49 16
= 784
(x')2 49x'y' + (y')2 8(x')2 16x'y' 8(y')2
= 784
49(x')2 98x'y' + 49(y')
2 16(x')
2 32x'y' 16(y')
2 = 1568
33(x)2 130xy + 33(y )2 1568 = 0
ANSWER:
33(x)2 130xy + 33(y )2 1568 = 0; hyperbola
9.4x2 25y2 = 64, =90
SOLUTION:
4x2 25y2 = 64, =90
Find the equations for x and y .
x = xcos y sin x = y y = xsin +y cos y = x Substitute into the original equation.
B2 4AC = 400 which is greater than 0, so the conic is a hyperbola.
4(y')2 25(x')
2 = 64
4(y')2 25(x')2 = 64
25(x)2 4(y )
2 + 64 = 0
ANSWER:
25(x)2 4(y )
2 + 64 = 0; hyperbola
10.6x2 + 5y2 = 30, =30
SOLUTION:
6x2 + 5y
2 = 30, =30
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 1920 which is less than 0 and B0,sotheconicisanellipse.
6 + 5
= 30
6 + 5
= 30
(x')2 3 x'y' + (y')
2 + (x')
2 + x'y' + (y')
2
= 30
18(x')2 12 x'y' + 6(y')
2 + 5(x')
2 + 10 x'y' + 15(y')
2 = 120
23(x)2 2 xy + 21(y )
2 120 = 0
ANSWER:
23(x)2 2 xy + 21(y )2 120 = 0; ellipse
Using a suitable angle of rotation for the conic with each given equation, write the equation in standard form.
11.xy = 4
SOLUTION:xy = 4
B2 4AC = 1 which is greater than 0, so the conic is a hyperbola.
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
cot 2 = 0
2 =
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
xy = 4
= 4
(x)2 xy + xy (y)
2 = 4
(x)2 (y)
2 = 4
(x)2 (y)
2 = 1
= 1
ANSWER:
=1
12.x2 xy + y2 = 2
SOLUTION:
x2 xy + y
2 = 2
B2 4AC = 3 which is less than 0 and B0,sotheconicisanellipse.
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
cot 2 = 0
2 =
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
x2 xy + y
2 = 2
+
= 2
(y)2
= 2
(x)2 + (y)
2
= 2
(x)2 + (y)
2 = 1
=1
ANSWER:
13.145x2 + 120xy + 180y2 = 900
SOLUTION:
145x2 + 120xy + 180y
2 = 900
B2 4AC = 90,000 which is less than 0 and B0,sotheconicisanellipse.
72 + 24
2 = 25
2
sin 2 = andcos2 =
Use the halfangle identities to determine sin andcos .
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
cot 2 =
sin =
cos =
=
=
= =
=
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
145x2 + 120xy + 180y
2 = 900
145 + 120 +180 = 900
145 + 120 +
180
= 900
(x)2 xy + (y)2 + (x)2 xy (y)2 + (x)2 + xy +
(y)2
= 900
261(x)2 696xy + 464(y)
2 + 288(x)
2 168xy 288(y)
2 + 576(x)
2 + 864xy + 324(y)
2 = 4500
1125(x)2 + 500(y)2 = 4500
+
= 1
ANSWER:
+ =1
14.16x2 24xy + 9y2 5x 90y + 25 = 0
SOLUTION:
16x2 24xy + 9y2 5x 90y + 25 = 0
B2 4AC = 0, so the conic is a parabola.
72 + 24
2 = 25
2
sin 2 = andcos2 =
Use the halfangle identities to determine sin andcos .
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
cot 2 =
sin =
cos =
=
=
= =
=
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
16x2 24xy + 9y
2 5x 90y + 25 = 0
16 24 + 9
5 90 + 25
= 0
16 24 +
9 3x + 4y 72x 54y + 25
= 0
(x)2 xy + (y)
2 (x)
2 + xy + (y)
2 + (x)
2
+
xy + (y)2 75x 50y + 25
= 0
144(x)2 384xy + 256(y)2 288(x)2 + 168xy + 288(y)2 + 144(x)2 +
216xy + 81(y)2 1875x 1250y + 625 = 0
625(y)2 1875x 1250y + 625 = 0
(y)2 2y + 1 3x = 0
(y 1)2 = 3x
ANSWER:
(y 1)2 = 3x
15.2x2 72xy + 23y2 + 100x 50y = 0
SOLUTION:
2x2 72xy + 23y
2 + 100x 50y = 0
B2 4AC = 5000 which is greater than 0, so the conic is a hyperbola.
72 + 24
2 = 25
2
sin 2 = andcos2 =
Use the halfangle identities to determine sin andcos .
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
cot 2 =
sin =
cos =
=
=
=
=
=
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
2x2 72xy + 23y2 + 100x 50y = 0
2 72 + 23 + 100
50
= 0
2 72
+23 + 80x 60y 30x 40y
= 0
(x)2 xy + (y)2 (x)2 xy + (y)2 + (x)2 + xy
+
(y)2 + 50x 100y
= 0
32(x)2 48xy + 18(y)
2 864(x)
2 504xy + 864 (y)
2 + 207(x)
2 + 552xy +
368(y)2 + 1250x 2500y = 0
625(x)2 + 1250(y)
2 + 1250x 2500y = 0
(x)2 + 2(y)2 + 2x 4y = 0
2(y)2 4y (x)
2 + 2x = 0
2[(y)2 2y + 1] [(x)2 2x + 1] 2+1 = 0
2(y 1)2 (x 1)
2 = 1
= 1
ANSWER:
16.x2 3y2 8x + 30y = 60
SOLUTION:
x2 3y
2 8x + 30y = 60
B2 4AC = 12 which is greater than 0, so the conic is a hyperbola.
The conic has not been rotated. Convert the equation to standard form.
cot 2 =
cot 2 =
tan 2 = or0
sin 2 = 0
2 = 0 = 0
x2 3y
2 8x + 30y = 60
x2 8x 3y2 + 30y 60 = 0
(x2 8x + 16) 3(y
2 10y + 25) 16 + 3(25) 60 = 0
(x 4)2 3(y 5)2 = 1
= 1
ANSWER:
17.8x2 + 12xy + 3y2 + 4 = 6
SOLUTION:
8x2 + 12xy + 3y
2 + 4 = 6
B2 4AC = 4 which is greater than 0, so the conic is a hyperbola.
=13
sin 2 = andcos2 =
Use the halfangle identities to determine sin andcos .
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
cot 2 =
sin =
cos =
= =
=
=
=
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
8x2 + 12xy + 3y
2 + 4 = 6
8 + 12 + 3
= 2
8 + 12
+ 3
= 2
(x)2 xy + (y)2 + (x)2 + xy (y)2 + (x)2 + xy + (y)2
= 2
72 (x)2 96xy + 32(y)
2 + 72 (x)
2 + 60xy 72 (y)
2 + 12(x)
2 + 36xy + 27(y)
2 = 26
156(x)2 13(y)2 = 26
6(x)2
= 1
= 1
ANSWER:
18.73x2 + 72xy + 52y2 + 25x + 50y 75 = 0
SOLUTION:
73x2 + 72xy + 52y
2 + 25x + 50y 75 = 0
B2 4AC = 10,000 which is less than 0 and B0,sotheconicisanellipse.
72 + 24
2 = 25
2
sin 2 = andcos2 =
Use the halfangle identities to determine sin andcos .
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
=
=
sin =
cos =
=
=
=
=
=
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
73x2 + 72xy + 52y
2 + 25x + 50y 75 = 0
73 + 72 + 52 +
25 + 50 75
= 0
73 + 72 +
52 + 20x 15y + 30x + 40y 75
= 0
(x)2 xy + (y)
2 + (x)
2 + xy (y)
2 +
(x)2 +
xy + (y)2 + 50x + 25y 75
= 0
1168(x)2 1752xy + 657(y)
2 + 864(x)
2 + 504xy 864 (y)
2 + 468(x)
2 +
1248xy + 832(y)2 + 1250x + 625y 1875 = 0
2500(x)2 + 625(y)
2 + 1250x + 625y 1875 = 0
4(x)2 + (y)2 + 2x + y 3 = 0
+(y)2 + y 3
= 0
+ (y)2 + y + 3
= 0
+
=
+
= 1
=1
ANSWER:
=1
Write an equation for each conic in the xyplane for the given equation in xy form and the given value of .
19.(x)2 + 3(y )2 = 8, =
SOLUTION:
(x)2 + 3(y )2 = 8, =
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
(x)2 + 3(y )
2 = 8
+ 3
= 8
x2 + xy + y
2 + 3
= 8
x2 + xy + y
2 + y
2 3xy + x
2 = 8
2x2 2xy + 2y2 = 8
x2 xy + y
2 4 = 0
ANSWER:
x2 xy + y2 4 = 0
20. =1, =
SOLUTION:
=1, =
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
= 1
9(x)2 (y)
2 = 225
9
= 225
9 y2 + xy x
2 = 225
x2 + 9xy + y
2 y
2 + xy x
2 = 225
4x2 + 10xy + 4y
2 225 = 0
ANSWER:
4x2 + 10xy + 4y
2 225 = 0
21. =1, =
SOLUTION:
=1, =
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
= 1
4(x)2 (y)2 = 36
4
= 36
4 y2 + xy x2
= 36
x2 + 2 xy + 3y
2 y2 + xy x2
= 36
4x2 + 8 xy + 12y
2 y2 + 2 xy 3x2 = 144
x2 + 10 xy + 11y
2 144 = 0
ANSWER:
x2 + 10 xy + 11y
2 144 = 0
22.(x)2 = 8(y ), =45
SOLUTION:
(x)2 = 8(y ), =45
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
(x)2 = 8(y)
(x)2 8y = 0
8
= 0
x2 + xy + y
2 4 y + 4 x
= 0
x2 + 2xy+y2 + 8 x 8 y = 0
ANSWER:
x2 + 2xy+y2 + 8 x 8 y = 0
23. + =1, =
SOLUTION:
+ =1, =
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
+
= 1
4(x)2 + (y)
2 = 28
4 +
= 28
4 + y2 xy + x
2
= 28
3x2 + 2 xy + y
2 + y
2 xy + x
2
= 28
12x2 + 8 xy + 4y
2 + 3y
2 2 xy + x
2 = 112
13x2 + 6 xy + 7y
2 112 = 0
ANSWER:
13x2 + 6 xy + 7y
2 112 = 0
24.4x = (y )2, =30
SOLUTION:
4x = (y )2, =30
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
4x = (y)2
4(x)2 (y)
2 = 0
4
= 0
2 x + 2y y2 + xy x
2
= 0
4(2 x + 2y y2 + xy x
2)
= (4)0
8 x 8y + 3y2 2 xy + x
2 = 0
x2 2 xy + 3y
2 8 x 8y = 0
ANSWER:
x2 2 xy + 3y
2 8 x 8y = 0
25. =1, =45
SOLUTION:
=1, =45
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
= 1
(x)2 4(y)2 = 64
4
= 64
x2 + xy + y
2 4
= 64
x2 + xy + y
2 2y2+ 4xy 2x2
= 64
x2 + 2xy + y
2 4y
2+ 8xy 4x
2 = 128
3x2 + 10xy 3y2 = 128
3x2 10xy + 3y
2 + 128 = 0
ANSWER:
3x2 10xy + 3y2 + 128 = 0
26.(x)2 = 5(y ), =
SOLUTION:
(x)2 = 5(y ), =
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
(x)2 = 5(y)
2
(x)2 5(y)2 = 0
5
= 0
x2 + xy + y
2 y + x
= 0
x2 + 2 xy + 3y
2 10y + 10 x = 0
x2 + 3y
2 + 2 xy + 10 x 10y = 0
ANSWER:
x2 + 3y
2 + 2 xy + 10 x 10y = 0
27. =1, =30
SOLUTION:
=1, =30
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
= 1
9(x)2 4(y)
2 = 36
9 4
= 36
9 4
= 36
x2 + xy + y
2 3y
2 + 2 xy x
2
= 36
27x2 + 18 xy + 9y
2 12y
2 + 8 xy 4x
2 = 144
23x2 + 26 xy 3y
2 144 = 0
ANSWER:
23x2 + 26 xy 3y
2 144 = 0
28. + =1, =60
SOLUTION:
+ =1, =60
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
+
= 1
4(x)2 + 3(y)2 = 12
4 + 3
= 12
4 + 3
= 12
x2 + 2 xy + 3y
2 + y
2 xy + x2
= 12
4x2 + 8 xy + 12y
2 + 3y
2 6 xy + 9x2 = 48
13x2 + 2 xy + 15y
2 48 = 0
ANSWER:
13x2 + 2 xy + 15y
2 48 = 0
29.ASTRONOMY Suppose 144(x)2 + 64(y )2= 576 models the shape in the xy plane of a reflecting mirror in a telescope. a.Ifthemirrorhasbeenrotated30,determinetheequationofthemirrorinthexyplane. b. Graph the equation.
SOLUTION:
a. Find the equations for x and y for =30.
x= x cos +y sin
x = x + y
y=y cos x sin
y = y x
Substitute these values into the original equation.
b. Graph the equation by solving for y .
84y2 + (40 x)y + (124x
2 576) = 0
Use the quadratic formula.
y =
y =
Graph the conic using your graphing calculator.
144(x)2 + 64(y )2 = 576
144 + 64
= 576
144 + 64
= 576
108x2 + 36y
2 + 72 xy + 48y
2 + 16x
2 32 xy = 576
124x2 + 40 xy + 84y
2 576 = 0
ANSWER:
a. 31x2 10 xy + 21y2 = 144
b.
Graph each equation if it has been rotated at the indicated angle.
30. + =160
SOLUTION:
+ =160
Find the equations for x and y .
x= x cos +y sin
x = x + y
y=y cos x sin
y = y x
Substitute these values into the original equation.
Graph the equation by solving for y .
31y2 + (10 x)y + 21x
2 144
Use the quadratic formula.
y =
y =
y =
Graph the conic using your graphing calculator.
+
= 1
9(x)2 + 4(y)
2 = 36
9 + 4
= 36
9 + 4
= 36
x2 + xy + y
2 + y
2 2 xy + 3x
2
= 36
9x2 + 18 xy + 27y
2 + 4y
2 8 xy + 12x
2 = 144
21x2 + 10 xy + 31y
2 144 = 0
ANSWER:
31. =1; 45
SOLUTION:
=1; 45
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
Graph the equation by solving for y .
7y2 (122x)y + 7x2 + 1800 = 0
Use the quadratic formula.
y =
y =
y =
Graph the conic using your graphing calculator.
= 1
36(x)2 25(y)2 = 900
36 25
= 900
36 25
= 900
9x2 + 36xy + 9y
2 y2 + 25xy x2
= 900
18x2 + 72xy + 18y
2 25y
2 + 50xy 25x
2 = 1800
7x2 + 122xy 7y2 1800 = 0
7x2 122xy + 7y
2 + 1800 = 0
ANSWER:
32.(x)2 + 6x y =9; 30
SOLUTION:
(x)2 + 6x y =9; 30
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.
x = x + y
y=y cos x sin
y = y x
Substitute these values into the original equation.
Graph the equation by solving for y .
y2 + (2 x + 12 2 )y + 3x
2 + (12 + 2)x + 36 = 0
Use the quadratic formula.
y =
y =
y =
Graph the conic using your graphing calculator.
(x)2 + 6x y = 9
+ 6 y + x
= 9
x2 + xy + y
2 + 3 x + 3y y + x
= 9
3x2 + 2 xy + y
2 + 12 x + 12y 2 y + 2x = 36
3x2 + 2 xy + y
2 + (12 + 2)x + (12 2 )y + 36 = 0
ANSWER:
33.8(x)2 + 6(y )2 = 24; 30
SOLUTION:
8(x)2 + 6(y )
2 = 24; 30
x = x + y
y=y cos x sin
y = y x
Substitute these values into the original equation.
Graph the equation by solving for y .
13y2 + (2 x) y + 15x
2 48 = 0
Use the quadratic formula.
y =
y =
y =
8(x)2 + 6(y )2 = 24
8 + 6
= 24
8 + 6
= 24
6x2 + 4 xy + 2y
2 + y
2 3 xy + x2
= 24
12x2 + 8 xy + 4y
2 + 9y
2 6 xy + 3x2 = 48
15x2 + 2 xy + 13y
2 48 = 0
ANSWER:
34. =1; 45
SOLUTION:
=1; 45
Use the rotation formulas for x and y to find the equation of the rotated conic in the xyplane.x= x cos +y sin
x = x + y
y = y cos x sin
y = y x
Substitute these values into the original equation.
Graph the equation by solving for y .
3y2 + (10x)y + 3x
2 32 = 0
y =
y =
y =
= 1
4(x)2 (y)
2 = 16
4
= 16
4 y2 + xy x
2 = 16
2x2 + 4xy +2y
2 y
2 + xy x
2 = 16
4x2 + 8xy + 4y
2 y2 + 2xy x2 = 32
3x2 + 10xy + 3y
2 32 = 0
ANSWER:
35.y = 3(x)2 2x+560
SOLUTION:
y = 3(x)2 2x+560
Find the equations for x and y .
x= x cos +y sin
x = x + y
y=y cos x sin
y = y x
Substitute these values into the original equation.
Graph the equation by solving for y .
9y2 + (2 + 4 6 x)y + (3x2 + (4 2 )x 20) = 0
y =
y =
y =
y = 3(x)2 2x+5
y 3(x)2 + 2x 5 = 0
y x 3 + 2 5
= 0
y x 3 + x + y 5
= 0
y x x2 xy y2 + x + y 5
= 0
2y 2 x 3x2 6 xy 9y2 + 4x + 4 y 20 = 0
3x2 6 xy 9y2 + (4 2 )x + (2 + 4 y 20 = 0
ANSWER:
36.COMMUNICATIONAsatellitedishtracksasatellitedirectlyoverhead.Supposey = x2 models the shape of
the dish when it is oriented in this position. Later in the day, the dish is observed to have rotated approximately 30. a. Write an equation that models the new orientation of the dish. b. Use a graphing calculator to graph both equations on the same screen. Sketch this graph on your paper.
SOLUTION:
a. y = x2, =30
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
b.
y = x2
x+ y
=
x+ y
=
x+ y
= (x)2 xy + (y)2
12x + 12 y = 3(x)2 2 xy + (y)2
0 = 3(x)2 2 xy + (y )2 12x 12 y
ANSWER:
a. 3(x)2 2 xy + (y )2 12x 12 y = 0
b.
GRAPHING CALCULATOR Graph the conic given by each equation.
37.x2 2xy + y2 5x 5y = 0
SOLUTION:Graph the equation by solving for y .
y =
y =
y =
x2 2xy + y2 5x 5y = 0
y2 + (2x 5)y + x
2 5x = 0
ANSWER:
38.2x2 + 9xy + 14y2 = 5
SOLUTION:Graph the equation by solving for y .
y =
y =
2x2 + 9xy + 14y
2 = 5
14y2 + (9x)y + 2x 5 = 0
ANSWER:
39.8x2 + 5xy 4y2 = 2
SOLUTION:Graph the equation by solving for y .
8x2 + 5xy 4y2 = 2
4y2 + (5x)y + 8x
2 + 2 = 0
y =
y =
y =
ANSWER:
40.2x2 + 4 xy + 6y2 + 3x = y
SOLUTION:Graph the equation by solving for y .
2x2 + 4 xy + 6y
2 + 3x = y
6y2 + (4 x 1)y + 2x
2 + 3x = 0
y =
y =
y =
ANSWER:
41.2x2 + 4xy + 2y2 + 2 x 2 y = 12
SOLUTION:Graph the equation by solving for y .
y =
y =
y =
2x2 + 4xy + 2y
2 + 2 x 2 y = 12
2y2 + (4x 2 )y + 2x
2 + 2 x + 12 = 0
ANSWER:
42.9x2 + 4xy + 6y2 = 20
SOLUTION:Graph the equation by solving for y .
y =
y =
y =
9x2 + 4xy + 6y
2 = 20
6y2 + (4x)y + 9x
2 20 = 0
ANSWER:
43.x2 + 10 xy + 11y2 64 = 0
SOLUTION:Graph the equation by solving for y .
y =
y =
x2 + 10 xy + 11y
2 64 = 0
11y2 + (10 x)y + x
2 64 = 0
ANSWER:
44.x2 + y2 4 = 0
SOLUTION:Graph the equation by solving for y .
y =
y =
y =
x2 + y
2 4 = 0
y2 + x
2 4 = 0
ANSWER:
45.x2 2 xy y2 + 18 = 0
SOLUTION:Graph the equation by solving for y .
x2 2 xy y
2 + 18 = 0
y2 (2 x)y + x
2 + 18 = 0
y =
y =
y = x
ANSWER:
46.2x2 + 9xy + 14y2 5 = 0
SOLUTION:Graph the equation by solving for y .
y =
y =
2x2 + 9xy + 14y
2 5 = 0
14y2 + (9x)y + 2x
2 5 = 0
eSolutions Manual - Powered by Cognero Page 3
7-4 Rotations of Conic Sections
-
Write each equation in the xy plane for the given value of . Then identify the conic.
1.x2 y2 = 9, =
SOLUTION:
x2 y2 = 9, =
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 16 which is greater than 0, so the conic is a hyperbola.
= 9
(x')2 x'y' x'y' + (y')2
= 9
(x')2 x'y' + (y')2 (x')2 x'y' (y')2
= 9
(x')2 x'y' + (y')
= 9
(x')2 2 x'y' + (y') = 18
(x)2 + 2 xy (y )2 + 18 = 0
ANSWER:
(x)2 + 2 xy (y )
2 + 18 = 0; hyperbola
2.xy = 8, =45
SOLUTION:
xy = 8, =45 Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 4 which is greater than 0, so the conic is a hyperbola.
= 8
(x')2 x'y' + x'y' (y')
2 = 8
(x')2 (y')
2 + 8
= 0
(x)2 (y )2 + 16 = 0
ANSWER:
(x)2 (y )2 + 16 = 0; hyperbola
3.x2 8y = 0, =
SOLUTION:
x2 8y = 0, =
Find the equations for x and y .
x = xcos y sin x = y y = xsin +y cos y = x Substitute into the original equation.
There is only one squared term, so the conic is a parabola.
(y')2 8x' = 0
(y')2 8x' = 0
ANSWER:
(y )2 8x = 0; parabola
4.2x2 + 2y2 = 8, =
SOLUTION:
2x2 + 2y
2 = 8, =
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 4 which is less than 0, B = 0, and A = C, so the conic is a circle.
2 + 2
= 8
2 + 2
= 8
(x')2 x'y' + (y)
2 + (x)
2 + x'y' + (y')
2 = 8
2(x)2 + 2(y )2 = 8
(x)2 + (y )
2 4 = 0
ANSWER:
(x)2 + (y )
2 4 = 0; circle
5.y2 + 8x = 0, =30
SOLUTION:
y2 + 8x = 0, =30
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 0, so the conic is a parabola.
+ 8
= 0
(x')2 + x'y' + x'y' + (y')
2 + 4 x' 4y'
= 0
(x')2 + x'y' + (y')
2 + 4 x' 4y'
= 0
(x)2 + 2 xy + 3(y )
2 + 16 x 16y' = 0
ANSWER:
(x)2 + 2 xy + 3(y )2 + 16 x 16y = 0; parabola
6.4x2 + 9y2 = 36, =30
SOLUTION:
4x2 + 9y
2 = 36, =30
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 2304 which is less than 0 and B0,sotheconicisanellipse.
4 + 9
= 36
4 + 9
= 36
3(x')2 2 x'y' + (y')2 + (x')2 + x'y' + (y')2
= 36
12(x')2 8 x'y' + 4(y')2 + 9(x')2 + 18 x'y' + 27(y')2 = 144
21(x)2 + 10 xy + 31(y )2 144 = 0
ANSWER:
21(x)2 + 10 xy + 31(y )
2 144 = 0; ellipse
7.x2 5x + y2 = 3, =45
SOLUTION:
x2 5x + y
2 = 3, =45
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 16 which is less than 0 and B0,sotheconicisanellipse.
5 +
= 3
(x')2 x'y' x'y' + (y')
2 x' + y' + (x')
2 + x'y' + x'y'
+ (y')2
= 3
(x')2 + (y')
2 x' + y' 3
= 0
2(x)2 + 2(y )
2 5 x + 5 y 6 = 0
ANSWER:
2(x)2 + 2(y )2 5 x + 5 y 6 = 0; ellipse
8.49x2 16y2 = 784, =
SOLUTION:
49x2 16y2 = 784, =
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 15,811 which is greater than 0, so the conic is a hyperbola.
49 16
= 784
49 16
= 784
(x')2 49x'y' + (y')2 8(x')2 16x'y' 8(y')2
= 784
49(x')2 98x'y' + 49(y')
2 16(x')
2 32x'y' 16(y')
2 = 1568
33(x)2 130xy + 33(y )2 1568 = 0
ANSWER:
33(x)2 130xy + 33(y )2 1568 = 0; hyperbola
9.4x2 25y2 = 64, =90
SOLUTION:
4x2 25y2 = 64, =90
Find the equations for x and y .
x = xcos y sin x = y y = xsin +y cos y = x Substitute into the original equation.
B2 4AC = 400 which is greater than 0, so the conic is a hyperbola.
4(y')2 25(x')
2 = 64
4(y')2 25(x')2 = 64
25(x)2 4(y )
2 + 64 = 0
ANSWER:
25(x)2 4(y )
2 + 64 = 0; hyperbola
10.6x2 + 5y2 = 30, =30
SOLUTION:
6x2 + 5y
2 = 30, =30
Find the equations for x and y .
x = xcos y sin
x = x y
y = xsin +y cos
y = x+ y
Substitute into the original equation.
B2 4AC = 1920 which is less than 0 and B0,sotheconicisanellipse.
6 + 5
= 30
6 + 5
= 30
(x')2 3 x'y' + (y')
2 + (x')
2 + x'y' + (y')
2
= 30
18(x')2 12 x'y' + 6(y')
2 + 5(x')
2 + 10 x'y' + 15(y')
2 = 120
23(x)2 2 xy + 21(y )
2 120 = 0
ANSWER:
23(x)2 2 xy + 21(y )2 120 = 0; ellipse
Using a suitable angle of rotation for the conic with each given equation, write the equation in standard form.
11.xy = 4
SOLUTION:xy = 4
B2 4AC = 1 which is greater than 0, so the conic is a hyperbola.
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
cot 2 = 0
2 =
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
xy = 4
= 4
(x)2 xy + xy (y)
2 = 4
(x)2 (y)
2 = 4
(x)2 (y)
2 = 1
= 1
ANSWER:
=1
12.x2 xy + y2 = 2
SOLUTION:
x2 xy + y
2 = 2
B2 4AC = 3 which is less than 0 and B0,sotheconicisanellipse.
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
cot 2 = 0
2 =
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
x2 xy + y
2 = 2
+
= 2
(y)2
= 2
(x)2 + (y)
2
= 2
(x)2 + (y)
2 = 1
=1
ANSWER:
13.145x2 + 120xy + 180y2 = 900
SOLUTION:
145x2 + 120xy + 180y
2 = 900
B2 4AC = 90,000 which is less than 0 and B0,sotheconicisanellipse.
72 + 24
2 = 25
2
sin 2 = andcos2 =
Use the halfangle identities to determine sin andcos .
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
cot 2 =
sin =
cos =
=
=
= =
=
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
145x2 + 120xy + 180y
2 = 900
145 + 120 +180 = 900
145 + 120 +
180
= 900
(x)2 xy + (y)2 + (x)2 xy (y)2 + (x)2 + xy +
(y)2
= 900
261(x)2 696xy + 464(y)
2 + 288(x)
2 168xy 288(y)
2 + 576(x)
2 + 864xy + 324(y)
2 = 4500
1125(x)2 + 500(y)2 = 4500
+
= 1
ANSWER:
+ =1
14.16x2 24xy + 9y2 5x 90y + 25 = 0
SOLUTION:
16x2 24xy + 9y2 5x 90y + 25 = 0
B2 4AC = 0, so the conic is a parabola.
72 + 24
2 = 25
2
sin 2 = andcos2 =
Use the halfangle identities to determine sin andcos .
Find the equations for x and y .
Substitute these values into the original equation.
cot 2 =
cot 2 =
sin =
cos =
=
=
= =
=
=
x = x cos y sin y = x sin +y cos
= x y
= x + y
16x2 24xy + 9y
2 5x 90y + 25 = 0
16 24 + 9
5 90 + 25
= 0
