 Algebra and Trigonometry 2e

Chapter 12

12.1The Ellipse

1 .

$x 2 + y 2 16 =1 x 2 + y 2 16 =1$

2 .

$( x−1 ) 2 16 + ( y−3 ) 2 4 =1 ( x−1 ) 2 16 + ( y−3 ) 2 4 =1$

3 .

center: $( 0,0 ); ( 0,0 );$ vertices: $( ±6,0 ); ( ±6,0 );$ co-vertices: $( 0,±2 ); ( 0,±2 );$ foci: $( ±4 2 ,0 ) ( ±4 2 ,0 )$ 4 .

Standard form: $x 2 16 + y 2 49 =1; x 2 16 + y 2 49 =1;$ center: $( 0,0 ); ( 0,0 );$ vertices: $( 0,±7 ); ( 0,±7 );$ co-vertices: $( ±4,0 ); ( ±4,0 );$ foci: $( 0,± 33 ) ( 0,± 33 )$ 5 .

Center: $( 4,2 ); ( 4,2 );$ vertices: $( −2,2 ) ( −2,2 )$ and $( 10,2 ); ( 10,2 );$ co-vertices: $( 4,2−2 5 ) ( 4,2−2 5 )$ and $( 4,2+2 5 ); ( 4,2+2 5 );$ foci: $( 0,2 ) ( 0,2 )$ and $( 8,2 ) ( 8,2 )$ 6 .

$(x−3) 2 4 + ( y+1 ) 2 16 =1; (x−3) 2 4 + ( y+1 ) 2 16 =1;$ center: $( 3,−1 ); ( 3,−1 );$ vertices: $( 3,−5 ) ( 3,−5 )$ and $( 3,3 ); ( 3,3 );$ co-vertices: $( 1,−1 ) ( 1,−1 )$ and $( 5,−1 ); ( 5,−1 );$ foci: $( 3,−1−2 3 ) ( 3,−1−2 3 )$ and $( 3,−1+2 3 ) ( 3,−1+2 3 )$

7 .
1. $x 2 57,600 + y 2 25,600 =1 x 2 57,600 + y 2 25,600 =1$
2. The people are standing 358 feet apart.

12.2The Hyperbola

1 .

Vertices: $( ±3,0 ); ( ±3,0 );$ Foci: $( ± 34 ,0 ) ( ± 34 ,0 )$

2 .

$y 2 4 − x 2 16 =1 y 2 4 − x 2 16 =1$

3 .

$( y−3 ) 2 25 + ( x−1 ) 2 144 =1 ( y−3 ) 2 25 + ( x−1 ) 2 144 =1$

4 .

vertices: $( ±12,0 ); ( ±12,0 );$ co-vertices: $( 0,±9 ); ( 0,±9 );$ foci: $( ±15,0 ); ( ±15,0 );$ asymptotes: $y=± 3 4 x; y=± 3 4 x;$ 5 .

center: $( 3,−4 ); ( 3,−4 );$ vertices: $( 3,−14 ) ( 3,−14 )$ and $( 3,6 ); ( 3,6 );$ co-vertices: $( −5,−4 ); ( −5,−4 );$ and $( 11,−4 ); ( 11,−4 );$ foci: $( 3,−4−2 41 ) ( 3,−4−2 41 )$ and $( 3,−4+2 41 ); ( 3,−4+2 41 );$ asymptotes: $y=± 5 4 ( x−3 )−4 y=± 5 4 ( x−3 )−4$ 6 .

The sides of the tower can be modeled by the hyperbolic equation.

12.3The Parabola

1 .

Focus: $( −4,0 ); ( −4,0 );$ Directrix: $x=4; x=4;$ Endpoints of the latus rectum: $( −4,±8 ) ( −4,±8 )$ 2 .

Focus: $( 0,2 ); ( 0,2 );$ Directrix: $y=−2; y=−2;$ Endpoints of the latus rectum: $( ±4,2 ). ( ±4,2 ).$ 3 .

$x 2 =14y. x 2 =14y.$

4 .

Vertex: $( 8,−1 ); ( 8,−1 );$ Axis of symmetry: $y=−1; y=−1;$ Focus: $( 9,−1 ); ( 9,−1 );$ Directrix: $x=7; x=7;$ Endpoints of the latus rectum: $( 9,−3 ) ( 9,−3 )$ and $( 9,1 ). ( 9,1 ).$ 5 .

Vertex: $( −2,3 ); ( −2,3 );$ Axis of symmetry: $x=−2; x=−2;$ Focus: $( −2,−2 ); ( −2,−2 );$ Directrix: $y=8; y=8;$ Endpoints of the latus rectum: $( −12,−2 ) ( −12,−2 )$ and $( 8,−2 ). ( 8,−2 ).$ 6 .
1. $y 2 =1280x y 2 =1280x$
2. The depth of the cooker is 500 mm

12.4Rotation of Axes

1 .
1. hyperbola
2. ellipse
2 .

$x ′ 2 4 + y ′ 2 1 =1 x ′ 2 4 + y ′ 2 1 =1$

3 .
1. hyperbola
2. ellipse

12.5Conic Sections in Polar Coordinates

1 .

ellipse; $e= 1 3 ;x=−2 e= 1 3 ;x=−2$

2 . 3 .

$r= 1 1−cosθ r= 1 1−cosθ$

4 .

$4−8x+3 x 2 − y 2 =0 4−8x+3 x 2 − y 2 =0$

12.1 Section Exercises

1 .

An ellipse is the set of all points in the plane the sum of whose distances from two fixed points, called the foci, is a constant.

3 .

This special case would be a circle.

5 .

It is symmetric about the x-axis, y-axis, and the origin.

7 .

yes; $x 2 3 2 + y 2 2 2 =1 x 2 3 2 + y 2 2 2 =1$

9 .

yes; $x 2 ( 1 2 ) 2 + y 2 ( 1 3 ) 2 =1 x 2 ( 1 2 ) 2 + y 2 ( 1 3 ) 2 =1$

11 .

$x 2 2 2 + y 2 7 2 =1; x 2 2 2 + y 2 7 2 =1;$ Endpoints of major axis $( 0,7 ) ( 0,7 )$ and $( 0,−7 ). ( 0,−7 ).$ Endpoints of minor axis $( 2,0 ) ( 2,0 )$ and $( −2,0 ). ( −2,0 ).$ Foci at $( 0,3 5 ),( 0,−3 5 ). ( 0,3 5 ),( 0,−3 5 ).$

13 .

$x 2 ( 1 ) 2 + y 2 ( 1 3 ) 2 =1; x 2 ( 1 ) 2 + y 2 ( 1 3 ) 2 =1;$ Endpoints of major axis $( 1,0 ) ( 1,0 )$ and $( −1,0 ). ( −1,0 ).$ Endpoints of minor axis $( 0, 1 3 ),( 0,− 1 3 ). ( 0, 1 3 ),( 0,− 1 3 ).$ Foci at $( 2 2 3 ,0 ),( − 2 2 3 ,0 ). ( 2 2 3 ,0 ),( − 2 2 3 ,0 ).$

15 .

$( x−2 ) 2 7 2 + ( y−4 ) 2 5 2 =1; ( x−2 ) 2 7 2 + ( y−4 ) 2 5 2 =1;$ Endpoints of major axis $( 9,4 ),( −5,4 ). ( 9,4 ),( −5,4 ).$ Endpoints of minor axis $( 2,9 ),( 2,−1 ). ( 2,9 ),( 2,−1 ).$ Foci at $( 2+2 6 ,4 ),( 2−2 6 ,4 ). ( 2+2 6 ,4 ),( 2−2 6 ,4 ).$

17 .

$( x+5 ) 2 2 2 + ( y−7 ) 2 3 2 =1; ( x+5 ) 2 2 2 + ( y−7 ) 2 3 2 =1;$ Endpoints of major axis $( −5,10 ),( −5,4 ). ( −5,10 ),( −5,4 ).$ Endpoints of minor axis $( −3,7 ),( −7,7 ). ( −3,7 ),( −7,7 ).$ Foci at $( −5,7+ 5 ),( −5,7− 5 ). ( −5,7+ 5 ),( −5,7− 5 ).$

19 .

$( x−1 ) 2 3 2 + ( y−4 ) 2 2 2 =1; ( x−1 ) 2 3 2 + ( y−4 ) 2 2 2 =1;$ Endpoints of major axis $( 4,4 ),( −2,4 ). ( 4,4 ),( −2,4 ).$ Endpoints of minor axis $( 1,6 ),( 1,2 ). ( 1,6 ),( 1,2 ).$ Foci at $( 1+ 5 ,4 ),( 1− 5 ,4 ). ( 1+ 5 ,4 ),( 1− 5 ,4 ).$

21 .

$( x−3 ) 2 ( 3 2 ) 2 + ( y−5 ) 2 ( 2 ) 2 =1; ( x−3 ) 2 ( 3 2 ) 2 + ( y−5 ) 2 ( 2 ) 2 =1;$ Endpoints of major axis $( 3+3 2 ,5 ),( 3−3 2 ,5 ). ( 3+3 2 ,5 ),( 3−3 2 ,5 ).$ Endpoints of minor axis $( 3,5+ 2 ),( 3,5− 2 ). ( 3,5+ 2 ),( 3,5− 2 ).$ Foci at $( 7,5 ),( −1,5 ). ( 7,5 ),( −1,5 ).$

23 .

$( x+5 ) 2 ( 5 ) 2 + ( y−2 ) 2 ( 2 ) 2 =1; ( x+5 ) 2 ( 5 ) 2 + ( y−2 ) 2 ( 2 ) 2 =1;$ Endpoints of major axis $( 0,2 ),( −10,2 ). ( 0,2 ),( −10,2 ).$ Endpoints of minor axis $( −5,4 ),( −5,0 ). ( −5,4 ),( −5,0 ).$ Foci at $( −5+ 21 ,2 ),( −5− 21 ,2 ). ( −5+ 21 ,2 ),( −5− 21 ,2 ).$

25 .

$( x+3 ) 2 ( 5 ) 2 + ( y+4 ) 2 ( 2 ) 2 =1; ( x+3 ) 2 ( 5 ) 2 + ( y+4 ) 2 ( 2 ) 2 =1;$ Endpoints of major axis $( 2,−4 ),( −8,−4 ). ( 2,−4 ),( −8,−4 ).$ Endpoints of minor axis $( −3,−2 ),( −3,−6 ). ( −3,−2 ),( −3,−6 ).$ Foci at $( −3+ 21 ,−4 ),( −3− 21 ,−4 ). ( −3+ 21 ,−4 ),( −3− 21 ,−4 ).$

27 .

Foci $( −3,−1+ 11 ),( −3,−1− 11 ) ( −3,−1+ 11 ),( −3,−1− 11 )$

29 .

Focus $( 0,0 ) ( 0,0 )$

31 .

Foci $( −10,30 ),( −10,−30 ) ( −10,30 ),( −10,−30 )$

33 .

Center $( 0,0 ), ( 0,0 ),$ Vertices $( 4,0 ),( −4,0 ),(0,3),(0,−3), ( 4,0 ),( −4,0 ),(0,3),(0,−3),$ Foci $( 7 ,0 ),( − 7 ,0 ) ( 7 ,0 ),( − 7 ,0 )$ 35 .

Center $( 0,0 ), ( 0,0 ),$ Vertices $( 1 9 ,0 ),( − 1 9 ,0 ),( 0, 1 7 ),( 0,− 1 7 ), ( 1 9 ,0 ),( − 1 9 ,0 ),( 0, 1 7 ),( 0,− 1 7 ),$ Foci $( 0, 4 2 63 ),( 0,− 4 2 63 ) ( 0, 4 2 63 ),( 0,− 4 2 63 )$ 37 .

Center $( −3,3 ), ( −3,3 ),$ Vertices $( 0,3 ),( −6,3 ),( −3,0 ),( −3,6 ), ( 0,3 ),( −6,3 ),( −3,0 ),( −3,6 ),$ Focus $( −3,3 ) ( −3,3 )$

Note that this ellipse is a circle. The circle has only one focus, which coincides with the center. 39 .

Center $( 1,1 ), ( 1,1 ),$ Vertices $( 5,1 ),( −3,1 ),( 1,3 ),( 1,−1 ), ( 5,1 ),( −3,1 ),( 1,3 ),( 1,−1 ),$ Foci $( 1,1+2 3 ),( 1,1−2 3 ) ( 1,1+2 3 ),( 1,1−2 3 )$ 41 .

Center $( −4,5 ), ( −4,5 ),$ Vertices $( −2,5 ),( −6,4 ),( −4,6 ),( −4,4 ), ( −2,5 ),( −6,4 ),( −4,6 ),( −4,4 ),$ Foci $( −4+ 3 ,5 ),( −4− 3 ,5 ) ( −4+ 3 ,5 ),( −4− 3 ,5 )$ 43 .

Center $( −2,1 ), ( −2,1 ),$ Vertices $( 0,1 ),( −4,1 ),( −2,5 ),( −2,−3 ), ( 0,1 ),( −4,1 ),( −2,5 ),( −2,−3 ),$ Foci $( −2,1+2 3 ),( −2,1−2 3 ) ( −2,1+2 3 ),( −2,1−2 3 )$ 45 .

Center $( −2,−2 ), ( −2,−2 ),$ Vertices $( 0,−2 ),( −4,−2 ),( −2,0 ),( −2,−4 ), ( 0,−2 ),( −4,−2 ),( −2,0 ),( −2,−4 ),$ Focus $( −2,−2 ) ( −2,−2 )$ 47 .

$x 2 25 + y 2 29 =1 x 2 25 + y 2 29 =1$

49 .

$( x−4 ) 2 25 + ( y−2 ) 2 1 =1 ( x−4 ) 2 25 + ( y−2 ) 2 1 =1$

51 .

$( x+3 ) 2 16 + ( y−4 ) 2 4 =1 ( x+3 ) 2 16 + ( y−4 ) 2 4 =1$

53 .

$x 2 81 + y 2 9 =1 x 2 81 + y 2 9 =1$

55 .

$( x+2 ) 2 4 + ( y−2 ) 2 9 =1 ( x+2 ) 2 4 + ( y−2 ) 2 9 =1$

57 .

59 .

square units.

61 .

square units.

63 .

$x 2 4 h 2 + y 2 1 4 h 2 =1 x 2 4 h 2 + y 2 1 4 h 2 =1$

65 .

$x 2 400 + y 2 144 =1 x 2 400 + y 2 144 =1$ . Distance = 17.32 feet

67 .

Approximately 51.96 feet

12.2 Section Exercises

1 .

A hyperbola is the set of points in a plane the difference of whose distances from two fixed points (foci) is a positive constant.

3 .

The foci must lie on the transverse axis and be in the interior of the hyperbola.

5 .

The center must be the midpoint of the line segment joining the foci.

7 .

yes $x 2 6 2 − y 2 3 2 =1 x 2 6 2 − y 2 3 2 =1$

9 .

yes $x 2 4 2 − y 2 5 2 =1 x 2 4 2 − y 2 5 2 =1$

11 .

$x 2 5 2 − y 2 6 2 =1; x 2 5 2 − y 2 6 2 =1;$ vertices: $( 5,0 ),( −5,0 ); ( 5,0 ),( −5,0 );$ foci: $( 61 ,0 ),( − 61 ,0 ); ( 61 ,0 ),( − 61 ,0 );$ asymptotes: $y= 6 5 x,y=− 6 5 x y= 6 5 x,y=− 6 5 x$

13 .

$y 2 2 2 − x 2 9 2 =1; y 2 2 2 − x 2 9 2 =1;$ vertices: $( 0,2 ),( 0,−2 ); ( 0,2 ),( 0,−2 );$ foci: $( 0, 85 ),( 0,− 85 ); ( 0, 85 ),( 0,− 85 );$ asymptotes: $y= 2 9 x,y=− 2 9 x y= 2 9 x,y=− 2 9 x$

15 .

$( x−1 ) 2 3 2 − ( y−2 ) 2 4 2 =1; ( x−1 ) 2 3 2 − ( y−2 ) 2 4 2 =1;$ vertices: $( 4,2 ),( −2,2 ); ( 4,2 ),( −2,2 );$ foci: $( 6,2 ),( −4,2 ); ( 6,2 ),( −4,2 );$ asymptotes: $y= 4 3 ( x−1 )+2,y=− 4 3 ( x−1 )+2 y= 4 3 ( x−1 )+2,y=− 4 3 ( x−1 )+2$

17 .

$( x−2 ) 2 7 2 − ( y+7 ) 2 7 2 =1; ( x−2 ) 2 7 2 − ( y+7 ) 2 7 2 =1;$ vertices: $( 9,−7 ),( −5,−7 ); ( 9,−7 ),( −5,−7 );$ foci: $( 2+7 2 ,−7 ),( 2−7 2 ,−7 ); ( 2+7 2 ,−7 ),( 2−7 2 ,−7 );$ asymptotes: $y=x−9,y=−x−5 y=x−9,y=−x−5$

19 .

$( x+3 ) 2 3 2 − ( y−3 ) 2 3 2 =1; ( x+3 ) 2 3 2 − ( y−3 ) 2 3 2 =1;$ vertices: $( 0,3 ),( −6,3 ); ( 0,3 ),( −6,3 );$ foci: $( −3+3 2 ,1 ),( −3−3 2 ,1 ); ( −3+3 2 ,1 ),( −3−3 2 ,1 );$ asymptotes: $y=x+6,y=−x y=x+6,y=−x$

21 .

$( y−4 ) 2 2 2 − ( x−3 ) 2 4 2 =1; ( y−4 ) 2 2 2 − ( x−3 ) 2 4 2 =1;$ vertices: $( 3,6 ),( 3,2 ); ( 3,6 ),( 3,2 );$ foci: $( 3,4+2 5 ),( 3,4−2 5 ); ( 3,4+2 5 ),( 3,4−2 5 );$ asymptotes: $y= 1 2 ( x−3 )+4,y=− 1 2 ( x−3 )+4 y= 1 2 ( x−3 )+4,y=− 1 2 ( x−3 )+4$

23 .

$( y+5 ) 2 7 2 − ( x+1 ) 2 70 2 =1; ( y+5 ) 2 7 2 − ( x+1 ) 2 70 2 =1;$ vertices: $( −1,2 ),( −1,−12 ); ( −1,2 ),( −1,−12 );$ foci: $( −1,−5+7 101 ),( −1,−5−7 101 ); ( −1,−5+7 101 ),( −1,−5−7 101 );$ asymptotes: $y= 1 10 ( x+1 )−5,y=− 1 10 ( x+1 )−5 y= 1 10 ( x+1 )−5,y=− 1 10 ( x+1 )−5$

25 .

$( x+3 ) 2 5 2 − ( y−4 ) 2 2 2 =1; ( x+3 ) 2 5 2 − ( y−4 ) 2 2 2 =1;$ vertices: $( 2,4 ),( −8,4 ); ( 2,4 ),( −8,4 );$ foci: $( −3+ 29 ,4 ),( −3− 29 ,4 ); ( −3+ 29 ,4 ),( −3− 29 ,4 );$ asymptotes: $y= 2 5 ( x+3 )+4,y=− 2 5 ( x+3 )+4 y= 2 5 ( x+3 )+4,y=− 2 5 ( x+3 )+4$

27 .

$y= 2 5 ( x−3 )−4,y=− 2 5 ( x−3 )−4 y= 2 5 ( x−3 )−4,y=− 2 5 ( x−3 )−4$

29 .

$y= 3 4 ( x−1 )+1,y=− 3 4 ( x−1 )+1 y= 3 4 ( x−1 )+1,y=− 3 4 ( x−1 )+1$

31 . 33 . 35 . 37 . 39 . 41 . 43 . 45 .

$x 2 9 − y 2 16 =1 x 2 9 − y 2 16 =1$

47 .

$( x−6 ) 2 25 − ( y−1 ) 2 11 =1 ( x−6 ) 2 25 − ( y−1 ) 2 11 =1$

49 .

$( x−4 ) 2 25 − ( y−2 ) 2 1 =1 ( x−4 ) 2 25 − ( y−2 ) 2 1 =1$

51 .

$y 2 16 − x 2 25 =1 y 2 16 − x 2 25 =1$

53 .

$y 2 9 − ( x+1 ) 2 9 =1 y 2 9 − ( x+1 ) 2 9 =1$

55 .

$( x+3 ) 2 25 − ( y+3 ) 2 25 =1 ( x+3 ) 2 25 − ( y+3 ) 2 25 =1$

57 .

$y( x )=3 x 2 +1 ,y( x )=−3 x 2 +1 y( x )=3 x 2 +1 ,y( x )=−3 x 2 +1$ 59 .

$y( x )=1+2 x 2 +4x+5 ,y( x )=1−2 x 2 +4x+5 y( x )=1+2 x 2 +4x+5 ,y( x )=1−2 x 2 +4x+5$ 61 .

$x 2 25 − y 2 25 =1 x 2 25 − y 2 25 =1$ 63 .

$x 2 100 − y 2 25 =1 x 2 100 − y 2 25 =1$ 65 .

$x 2 400 − y 2 225 =1 x 2 400 − y 2 225 =1$ 67 .

$4(x-1)2-y22=16 4(x-1)2-y22=16$

69 .

$( x−h ) 2 a2 =4 - (y-k)2 b2 =(x-3)2-9y2=4 ( x−h ) 2 a2 =4 -(y-k)2 b2=(x-3)2-9y2=4$

12.3 Section Exercises

1 .

A parabola is the set of points in the plane that lie equidistant from a fixed point, the focus, and a fixed line, the directrix.

3 .

The graph will open down.

5 .

The distance between the focus and directrix will increase.

7 .

yes $x2=4(116)y x2=4(116)y$

9 .

yes $( y−3 ) 2 =4(2)( x−2 ) ( y−3 ) 2 =4(2)( x−2 )$

11 .

$y 2 = 1 8 x,V:(0,0);F:( 1 32 ,0 );d:x=− 1 32 y 2 = 1 8 x,V:(0,0);F:( 1 32 ,0 );d:x=− 1 32$

13 .

$x 2 =− 1 4 y,V:( 0,0 );F:( 0,− 1 16 );d:y= 1 16 x 2 =− 1 4 y,V:( 0,0 );F:( 0,− 1 16 );d:y= 1 16$

15 .

$y 2 = 1 36 x,V:( 0,0 );F:( 1 144 ,0 );d:x=− 1 144 y 2 = 1 36 x,V:( 0,0 );F:( 1 144 ,0 );d:x=− 1 144$

17 .

$( x−1 ) 2 =4( y−1 ),V:( 1,1 );F:( 1,2 );d:y=0 ( x−1 ) 2 =4( y−1 ),V:( 1,1 );F:( 1,2 );d:y=0$

19 .

$( y−4 ) 2 =2( x+3 ),V:( −3,4 );F:( − 5 2 ,4 );d:x=− 7 2 ( y−4 ) 2 =2( x+3 ),V:( −3,4 );F:( − 5 2 ,4 );d:x=− 7 2$

21 .

$( x+4 ) 2 =24( y+1 ),V:( −4,−1 );F:( −4,5 );d:y=−7 ( x+4 ) 2 =24( y+1 ),V:( −4,−1 );F:( −4,5 );d:y=−7$

23 .

$( y−3 ) 2 =−12( x+1 ),V:( −1,3 );F:( −4,3 );d:x=2 ( y−3 ) 2 =−12( x+1 ),V:( −1,3 );F:( −4,3 );d:x=2$

25 .

$( x−5 ) 2 = 4 5 ( y+3 ),V:( 5,−3 );F:( 5,− 14 5 );d:y=− 16 5 ( x−5 ) 2 = 4 5 ( y+3 ),V:( 5,−3 );F:( 5,− 14 5 );d:y=− 16 5$

27 .

$( x−2 ) 2 =−2( y−5 ),V:( 2,5 );F:( 2, 9 2 );d:y= 11 2 ( x−2 ) 2 =−2( y−5 ),V:( 2,5 );F:( 2, 9 2 );d:y= 11 2$

29 .

$( y−1 ) 2 = 4 3 ( x−5 ),V:( 5,1 );F:( 16 3 ,1 );d:x= 14 3 ( y−1 ) 2 = 4 3 ( x−5 ),V:( 5,1 );F:( 16 3 ,1 );d:x= 14 3$

31 . 33 . 35 . 37 . 39 . 41 . 43 . 45 .

$x 2 =−16y x 2 =−16y$

47 .

$( y−2 ) 2 =4 2 ( x−2 ) ( y−2 ) 2 =4 2 ( x−2 )$

49 .

$( y+ 3 ) 2 =−4 2 ( x− 2 ) ( y+ 3 ) 2 =−4 2 ( x− 2 )$

51 .

$x 2 =y x 2 =y$

53 .

$( y−2 ) 2 = 1 4 ( x+2 ) ( y−2 ) 2 = 1 4 ( x+2 )$

55 .

$( y− 3 ) 2 =4 5 ( x+ 2 ) ( y− 3 ) 2 =4 5 ( x+ 2 )$

57 .

$y 2 =−8x y 2 =−8x$

59 .

$( y+1 ) 2 =12( x+3 ) ( y+1 ) 2 =12( x+3 )$

61 .

$( 0,1 ) ( 0,1 )$

63 .

At the point 2.25 feet above the vertex.

65 .

0.5625 feet

67 .

$x 2 =−125( y−20 ), x 2 =−125( y−20 ),$ height is 7.2 feet

69 .

2304 feet

12.4 Section Exercises

1 .

The $xy xy$ term causes a rotation of the graph to occur.

3 .

The conic section is a hyperbola.

5 .

It gives the angle of rotation of the axes in order to eliminate the $xy xy$ term.

7 .

$AB=0, AB=0,$ parabola

9 .

$AB=−4<0, AB=−4<0,$ hyperbola

11 .

$AB=6>0, AB=6>0,$ ellipse

13 .

$B 2 −4AC=0, B 2 −4AC=0,$ parabola

15 .

$B 2 −4AC=0, B 2 −4AC=0,$ parabola

17 .

$B 2 −4AC=−96<0, B 2 −4AC=−96<0,$ ellipse

19 .

$7 x ′ 2 +9 y ′ 2 −4=0 7 x ′ 2 +9 y ′ 2 −4=0$

21 .

$3 x ′ 2 +2 x ′ y ′ −5 y ′ 2 +1=0 3 x ′ 2 +2 x ′ y ′ −5 y ′ 2 +1=0$

23 .

$θ= 60 ∘ ,11 x ′ 2 − y ′ 2 + 3 x ′ + y ′ −4=0 θ= 60 ∘ ,11 x ′ 2 − y ′ 2 + 3 x ′ + y ′ −4=0$

25 .

$θ= 150 ∘ ,21 x ′ 2 +9 y ′ 2 +4 x ′ −4 3 y ′ −6=0 θ= 150 ∘ ,21 x ′ 2 +9 y ′ 2 +4 x ′ −4 3 y ′ −6=0$

27 .

$θ≈ 36.9 ∘ ,125 x ′ 2 +6 x ′ −42 y ′ +10=0 θ≈ 36.9 ∘ ,125 x ′ 2 +6 x ′ −42 y ′ +10=0$

29 .

$θ= 45 ∘ ,3 x ′ 2 − y ′ 2 − 2 x ′ + 2 y ′ +1=0 θ= 45 ∘ ,3 x ′ 2 − y ′ 2 − 2 x ′ + 2 y ′ +1=0$

31 .

$2 2 ( x ′ + y ′ )= 1 2 ( x ′ − y ′ ) 2 2 2 ( x ′ + y ′ )= 1 2 ( x ′ − y ′ ) 2$ 33 .

$( x ′ − y ′ ) 2 8 + ( x ′ + y ′ ) 2 2 =1 ( x ′ − y ′ ) 2 8 + ( x ′ + y ′ ) 2 2 =1$ 35 .

$( x ′ + y ′ ) 2 2 − ( x ′ − y ′ ) 2 2 =1 ( x ′ + y ′ ) 2 2 − ( x ′ − y ′ ) 2 2 =1$ 37 .

$3 2 x ′ − 1 2 y ′ = ( 1 2 x ′ + 3 2 y ′ −1 ) 2 3 2 x ′ − 1 2 y ′ = ( 1 2 x ′ + 3 2 y ′ −1 ) 2$ 39 . 41 . 43 . 45 . 47 . 49 . 51 .

$θ= 45 ∘ θ= 45 ∘$ 53 .

$θ= 60 ∘ θ= 60 ∘$ 55 .

$θ≈ 36.9 ∘ θ≈ 36.9 ∘$ 57 .

$−4 6

59 .

$k=2 k=2$

12.5 Section Exercises

1 .

If eccentricity is less than 1, it is an ellipse. If eccentricity is equal to 1, it is a parabola. If eccentricity is greater than 1, it is a hyperbola.

3 .

The directrix will be parallel to the polar axis.

5 .

One of the foci will be located at the origin.

7 .

Parabola with $e=1 e=1$ and directrix $3 4 3 4$ units below the pole.

9 .

Hyperbola with $e=2 e=2$ and directrix $5 2 5 2$ units above the pole.

11 .

Parabola with $e=1 e=1$ and directrix $3 10 3 10$ units to the right of the pole.

13 .

Ellipse with $e= 2 7 e= 2 7$ and directrix $2 2$ units to the right of the pole.

15 .

Hyperbola with $e= 5 3 e= 5 3$ and directrix $11 5 11 5$ units above the pole.

17 .

Hyperbola with $e= 8 7 e= 8 7$ and directrix $7 8 7 8$ units to the right of the pole.

19 .

$25 x 2 +16 y 2 −12y−4=0 25 x 2 +16 y 2 −12y−4=0$

21 .

$21 x 2 −4 y 2 −30x+9=0 21 x 2 −4 y 2 −30x+9=0$

23 .

$64 y 2 =48x+9 64 y 2 =48x+9$

25 .

$96 y 2 −25 x 2 +110y+25=0 96 y 2 −25 x 2 +110y+25=0$

27 .

$3 x 2 +4 y 2 −2x−1=0 3 x 2 +4 y 2 −2x−1=0$

29 .

$5 x 2 +9 y 2 −24x−36=0 5 x 2 +9 y 2 −24x−36=0$

31 . 33 . 35 . 37 . 39 . 41 . 43 .

$r= 4 5+cosθ r= 4 5+cosθ$

45 .

$r= 4 1+2sinθ r= 4 1+2sinθ$

47 .

$r= 1 1+cosθ r= 1 1+cosθ$

49 .

$r= 7 8−28cosθ r= 7 8−28cosθ$

51 .

$r= 12 2+3sinθ r= 12 2+3sinθ$

53 .

$r= 15 4−3cosθ r= 15 4−3cosθ$

55 .

$r= 3 3−3cosθ r= 3 3−3cosθ$

57 .

$r=± 2 1+sinθcosθ r=± 2 1+sinθcosθ$

59 .

$r=± 2 4cosθ+3sinθ r=± 2 4cosθ+3sinθ$

Review Exercises

1 .

$x 2 5 2 + y 2 8 2 =1; x 2 5 2 + y 2 8 2 =1;$ center: $( 0,0 ); ( 0,0 );$ vertices: $( 5,0 ),( −5,0 ),( 0,8 ),( 0,−8 ); ( 5,0 ),( −5,0 ),( 0,8 ),( 0,−8 );$ foci: $( 0, 39 ),( 0,− 39 ) ( 0, 39 ),( 0,− 39 )$

3 .

$(x+3) 2 1 2 + (y−2) 2 3 2 =1(−3,2);(−2,2),(−4,2),(−3,5),(−3,−1);( −3,2+2 2 ),( −3,2−2 2 ) (x+3) 2 1 2 + (y−2) 2 3 2 =1(−3,2);(−2,2),(−4,2),(−3,5),(−3,−1);( −3,2+2 2 ),( −3,2−2 2 )$

5 .

center: $( 0,0 ); ( 0,0 );$ vertices: $( 6,0 ),( −6,0 ),( 0,3 ),( 0,−3 ); ( 6,0 ),( −6,0 ),( 0,3 ),( 0,−3 );$ foci: $( 3 3 ,0 ),( −3 3 ,0 ) ( 3 3 ,0 ),( −3 3 ,0 )$ 7 .

center: $( −2,−2 ); ( −2,−2 );$ vertices: $( 2,−2 ),( −6,−2 ),( −2,6 ),( −2,−10 ); ( 2,−2 ),( −6,−2 ),( −2,6 ),( −2,−10 );$ foci: $( −2,−2+4 3 , ),( −2,−2−4 3 ) ( −2,−2+4 3 , ),( −2,−2−4 3 )$ 9 .

$x 2 25 + y 2 16 =1 x 2 25 + y 2 16 =1$

11 .

Approximately 35.71 feet

13 .

$( y+1 ) 2 4 2 − ( x−4 ) 2 6 2 =1; ( y+1 ) 2 4 2 − ( x−4 ) 2 6 2 =1;$ center: $( 4,−1 ); ( 4,−1 );$ vertices: $( 4,3 ),( 4,−5 ); ( 4,3 ),( 4,−5 );$ foci: $( 4,−1+2 13 ),( 4,−1−2 13 ) ( 4,−1+2 13 ),( 4,−1−2 13 )$

15 .

$( x−2 ) 2 2 2 − ( y+3 ) 2 ( 2 3 ) 2 =1; ( x−2 ) 2 2 2 − ( y+3 ) 2 ( 2 3 ) 2 =1;$ center: $( 2,−3 ); ( 2,−3 );$ vertices: $( 4,−3 ),( 0,−3 ); ( 4,−3 ),( 0,−3 );$ foci: $( 6,−3 ),( −2,−3 ) ( 6,−3 ),( −2,−3 )$

17 . 19 . 21 .

$( x−5 ) 2 1 − ( y−7 ) 2 3 =1 ( x−5 ) 2 1 − ( y−7 ) 2 3 =1$

23 .

$( x+2 ) 2 = 1 2 ( y−1 ); ( x+2 ) 2 = 1 2 ( y−1 );$ vertex: $( −2,1 ); ( −2,1 );$ focus: $( −2, 9 8 ); ( −2, 9 8 );$ directrix: $y= 7 8 y= 7 8$

25 .

$( x+5 ) 2 =( y+2 ); ( x+5 ) 2 =( y+2 );$ vertex: $( −5,−2 ); ( −5,−2 );$ focus: $( −5,− 7 4 ); ( −5,− 7 4 );$ directrix: $y=− 9 4 y=− 9 4$

27 . 29 . 31 .

$( x−2 ) 2 =( 1 2 )( y−1 ) ( x−2 ) 2 =( 1 2 )( y−1 )$

33 .

$B 2 −4AC=0, B 2 −4AC=0,$ parabola

35 .

$B 2 −4AC=−31<0, B 2 −4AC=−31<0,$ ellipse

37 .

$θ= 45 ∘ , x ′ 2 +3 y ′ 2 −12=0 θ= 45 ∘ , x ′ 2 +3 y ′ 2 −12=0$

39 .

$θ= 45 ∘ θ= 45 ∘$ 41 .

Hyperbola with $e=5 e=5$ and directrix $2 2$ units to the left of the pole.

43 .

Ellipse with $e= 3 4 e= 3 4$ and directrix $1 3 1 3$ unit above the pole.

45 . 47 . 49 .

Practice Test

1 .

$x 2 3 2 + y 2 2 2 =1; x 2 3 2 + y 2 2 2 =1;$ center: $( 0,0 ); ( 0,0 );$ vertices: $( 3,0 ),( –3,0 ),( 0,2 ),( 0,−2 ); ( 3,0 ),( –3,0 ),( 0,2 ),( 0,−2 );$ foci: $( 5 ,0 ),( − 5 ,0 ) ( 5 ,0 ),( − 5 ,0 )$

3 .

center: $( 3,2 ); ( 3,2 );$ vertices: $( 11,2 ),( −5,2 ),( 3,8 ),( 3,−4 ); ( 11,2 ),( −5,2 ),( 3,8 ),( 3,−4 );$ foci: $( 3+2 7 ,2 ),( 3−2 7 ,2 ) ( 3+2 7 ,2 ),( 3−2 7 ,2 )$ 5 .

$( x−1 ) 2 36 + ( y−2 ) 2 27 =1 ( x−1 ) 2 36 + ( y−2 ) 2 27 =1$

7 .

$x 2 7 2 − y 2 9 2 =1; x 2 7 2 − y 2 9 2 =1;$ center: $( 0,0 ); ( 0,0 );$ vertices $( 7,0 ),( −7,0 ); ( 7,0 ),( −7,0 );$ foci: $( 130 ,0 ),( − 130 ,0 ); ( 130 ,0 ),( − 130 ,0 );$ asymptotes: $y=± 9 7 x y=± 9 7 x$

9 .

center: $( 3,−3 ); ( 3,−3 );$ vertices: $( 8,−3 ),( −2,−3 ); ( 8,−3 ),( −2,−3 );$ foci: $( 3+ 26 ,−3 ),( 3− 26 ,−3 ); ( 3+ 26 ,−3 ),( 3− 26 ,−3 );$ asymptotes: $y=± 1 5 (x−3)−3 y=± 1 5 (x−3)−3$ 11 .

$( y−3 ) 2 1 − ( x−1 ) 2 8 =1 ( y−3 ) 2 1 − ( x−1 ) 2 8 =1$

13 .

$( x−2 ) 2 = 1 3 ( y+1 ); ( x−2 ) 2 = 1 3 ( y+1 );$ vertex: $( 2,−1 ); ( 2,−1 );$ focus: $( 2,− 11 12 ); ( 2,− 11 12 );$ directrix: $y=− 13 12 y=− 13 12$

15 . 17 .

Approximately $8.49 8.49$ feet

19 .

parabola; $θ≈ 63.4 ∘ θ≈ 63.4 ∘$

21 .

$x ′ 2 −4 x ′ +3 y ′ =0 x ′ 2 −4 x ′ +3 y ′ =0$ 23 .

Hyperbola with $e= 3 2 , e= 3 2 ,$ and directrix $5 6 5 6$ units to the right of the pole.

25 . 