Section 7.1 Solutions

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; $\frac{{x}^{2}}{{3}^{2}}+\frac{{y}^{2}}{{2}^{2}}=1$

9. yes; $\frac{{x}^{2}}{{\left(\frac{1}{2}\right)}^{2}}+\frac{{y}^{2}}{{\left(\frac{1}{3}\right)}^{2}}=1$

11. $\frac{{x}^{2}}{{2}^{2}}+\frac{{y}^{2}}{{7}^{2}}=1$; Endpoints of major axis $\left(0,7\right)$ and $\left(0,-7\right)$. Endpoints of minor axis $\left(2,0\right)$ and $\left(-2,0\right)$. Foci at $\left(0,3\sqrt{5}\right),\left(0,-3\sqrt{5}\right)$.

13. $\frac{{x}^{2}}{{\left(1\right)}^{2}}+\frac{{y}^{2}}{{\left(\frac{1}{3}\right)}^{2}}=1$; Endpoints of major axis $\left(1,0\right)$ and $\left(-1,0\right)$. Endpoints of minor axis $\left(0,\frac{1}{3}\right),\left(0,-\frac{1}{3}\right)$. Foci at $\left(\frac{2\sqrt{2}}{3},0\right),\left(-\frac{2\sqrt{2}}{3},0\right)$.

15. $\frac{{\left(x - 2\right)}^{2}}{{7}^{2}}+\frac{{\left(y - 4\right)}^{2}}{{5}^{2}}=1$; Endpoints of major axis $\left(9,4\right),\left(-5,4\right)$. Endpoints of minor axis $\left(2,9\right),\left(2,-1\right)$. Foci at $\left(2+2\sqrt{6},4\right),\left(2 - 2\sqrt{6},4\right)$.

17. $\frac{{\left(x+5\right)}^{2}}{{2}^{2}}+\frac{{\left(y - 7\right)}^{2}}{{3}^{2}}=1$; Endpoints of major axis $\left(-5,10\right),\left(-5,4\right)$. Endpoints of minor axis $\left(-3,7\right),\left(-7,7\right)$. Foci at $\left(-5,7+\sqrt{5}\right),\left(-5,7-\sqrt{5}\right)$.

19. $\frac{{\left(x - 1\right)}^{2}}{{3}^{2}}+\frac{{\left(y - 4\right)}^{2}}{{2}^{2}}=1$; Endpoints of major axis $\left(4,4\right),\left(-2,4\right)$. Endpoints of minor axis $\left(1,6\right),\left(1,2\right)$. Foci at $\left(1+\sqrt{5},4\right),\left(1-\sqrt{5},4\right)$.

21. $\frac{{\left(x - 3\right)}^{2}}{{\left(3\sqrt{2}\right)}^{2}}+\frac{{\left(y - 5\right)}^{2}}{{\left(\sqrt{2}\right)}^{2}}=1$; Endpoints of major axis $\left(3+3\sqrt{2},5\right),\left(3 - 3\sqrt{2},5\right)$. Endpoints of minor axis $\left(3,5+\sqrt{2}\right),\left(3,5-\sqrt{2}\right)$. Foci at $\left(7,5\right),\left(-1,5\right)$.

23. $\frac{{\left(x+5\right)}^{2}}{{\left(5\right)}^{2}}+\frac{{\left(y - 2\right)}^{2}}{{\left(2\right)}^{2}}=1$; Endpoints of major axis $\left(0,2\right),\left(-10,2\right)$. Endpoints of minor axis $\left(-5,4\right),\left(-5,0\right)$. Foci at $\left(-5+\sqrt{21},2\right),\left(-5-\sqrt{21},2\right)$.

25. $\frac{{\left(x+3\right)}^{2}}{{\left(5\right)}^{2}}+\frac{{\left(y+4\right)}^{2}}{{\left(2\right)}^{2}}=1$; Endpoints of major axis $\left(2,-4\right),\left(-8,-4\right)$. Endpoints of minor axis $\left(-3,-2\right),\left(-3,-6\right)$. Foci at $\left(-3+\sqrt{21},-4\right),\left(-3-\sqrt{21},-4\right)$.

27. Foci $\left(-3,-1+\sqrt{11}\right),\left(-3,-1-\sqrt{11}\right)$

29. Focus $\left(0,0\right)$

31. Foci $\left(-10,30\right),\left(-10,-30\right)$

33. Center $\left(0,0\right)$, Vertices $\left(4,0\right),\left(-4,0\right),\left(0,3\right),\left(0,-3\right)$, Foci $\left(\sqrt{7},0\right),\left(-\sqrt{7},0\right)$

35. Center $\left(0,0\right)$, Vertices $\left(\frac{1}{9},0\right),\left(-\frac{1}{9},0\right),\left(0,\frac{1}{7}\right),\left(0,-\frac{1}{7}\right)$, Foci $\left(0,\frac{4\sqrt{2}}{63}\right),\left(0,-\frac{4\sqrt{2}}{63}\right)$

37. Center $\left(-3,3\right)$, Vertices $\left(0,3\right),\left(-6,3\right),\left(-3,0\right),\left(-3,6\right)$, Focus $\left(-3,3\right)$
Note that this ellipse is a circle. The circle has only one focus, which coincides with the center.

39. Center $\left(1,1\right)$, Vertices $\left(5,1\right),\left(-3,1\right),\left(1,3\right),\left(1,-1\right)$, Foci $\left(1,1+4\sqrt{3}\right),\left(1,1 - 4\sqrt{3}\right)$

41. Center $\left(-4,5\right)$, Vertices $\left(-2,5\right),\left(-6,4\right),\left(-4,6\right),\left(-4,4\right)$, Foci $\left(-4+\sqrt{3},5\right),\left(-4-\sqrt{3},5\right)$

43. Center $\left(-2,1\right)$, Vertices $\left(0,1\right),\left(-4,1\right),\left(-2,5\right),\left(-2,-3\right)$, Foci $\left(-2,1+2\sqrt{3}\right),\left(-2,1 - 2\sqrt{3}\right)$

45. Center $\left(-2,-2\right)$, Vertices $\left(0,-2\right),\left(-4,-2\right),\left(-2,0\right),\left(-2,-4\right)$, Focus $\left(-2,-2\right)$

47. $\frac{{x}^{2}}{25}+\frac{{y}^{2}}{29}=1$

49. $\frac{{\left(x - 4\right)}^{2}}{25}+\frac{{\left(y - 2\right)}^{2}}{1}=1$

51. $\frac{{\left(x+3\right)}^{2}}{16}+\frac{{\left(y - 4\right)}^{2}}{4}=1$

53. $\frac{{x}^{2}}{81}+\frac{{y}^{2}}{9}=1$

55. $\frac{{\left(x+2\right)}^{2}}{4}+\frac{{\left(y - 2\right)}^{2}}{9}=1$

57. $\text{Area}=12\pi$ square units

59. $\text{Area}=2\sqrt{5}\pi$ square units

61. $\text{Area }9\pi$ square units

63. $\frac{{x}^{2}}{4{h}^{2}}+\frac{{y}^{2}}{\frac{1}{4}{h}^{2}}=1$

65. $\frac{{x}^{2}}{400}+\frac{{y}^{2}}{144}=1$. Distance = 17.32 feet

67. Approximately 51.96 feet

Section 7.2 Solutions

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 $\frac{{x}^{2}}{{6}^{2}}-\frac{{y}^{2}}{{3}^{2}}=1$

9. yes $\frac{{x}^{2}}{{4}^{2}}-\frac{{y}^{2}}{{5}^{2}}=1$

11. $\frac{{x}^{2}}{{5}^{2}}-\frac{{y}^{2}}{{6}^{2}}=1$; vertices: $\left(5,0\right),\left(-5,0\right)$; foci: $\left(\sqrt{61},0\right),\left(-\sqrt{61},0\right)$; asymptotes: $y=\frac{6}{5}x,y=-\frac{6}{5}x$

13. $\frac{{y}^{2}}{{2}^{2}}-\frac{{x}^{2}}{{9}^{2}}=1$; vertices: $\left(0,2\right),\left(0,-2\right)$; foci: $\left(0,\sqrt{85}\right),\left(0,-\sqrt{85}\right)$; asymptotes: $y=\frac{2}{9}x,y=-\frac{2}{9}x$

15. $\frac{{\left(x - 1\right)}^{2}}{{3}^{2}}-\frac{{\left(y - 2\right)}^{2}}{{4}^{2}}=1$; vertices: $\left(4,2\right),\left(-2,2\right)$; foci: $\left(6,2\right),\left(-4,2\right)$; asymptotes: $y=\frac{4}{3}\left(x - 1\right)+2,y=-\frac{4}{3}\left(x - 1\right)+2$

17. $\frac{{\left(x - 2\right)}^{2}}{{7}^{2}}-\frac{{\left(y+7\right)}^{2}}{{7}^{2}}=1$; vertices: $\left(9,-7\right),\left(-5,-7\right)$; foci: $\left(2+7\sqrt{2},-7\right),\left(2 - 7\sqrt{2},-7\right)$; asymptotes: $y=x - 9,y=-x - 5$

19. $\frac{{\left(x+3\right)}^{2}}{{3}^{2}}-\frac{{\left(y - 3\right)}^{2}}{{3}^{2}}=1$; vertices: $\left(0,3\right),\left(-6,3\right)$; foci: $\left(-3+3\sqrt{2},1\right),\left(-3 - 3\sqrt{2},1\right)$; asymptotes: $y=x+6,y=-x$

21. $\frac{{\left(y - 4\right)}^{2}}{{2}^{2}}-\frac{{\left(x - 3\right)}^{2}}{{4}^{2}}=1$; vertices: $\left(3,6\right),\left(3,2\right)$; foci: $\left(3,4+2\sqrt{5}\right),\left(3,4 - 2\sqrt{5}\right)$; asymptotes: $y=\frac{1}{2}\left(x - 3\right)+4,y=-\frac{1}{2}\left(x - 3\right)+4$

23. $\frac{{\left(y+5\right)}^{2}}{{7}^{2}}-\frac{{\left(x+1\right)}^{2}}{{70}^{2}}=1$; vertices: $\left(-1,2\right),\left(-1,-12\right)$; foci: $\left(-1,-5+7\sqrt{101}\right),\left(-1,-5 - 7\sqrt{101}\right)$; asymptotes: $y=\frac{1}{10}\left(x+1\right)-5,y=-\frac{1}{10}\left(x+1\right)-5$

25. $\frac{{\left(x+3\right)}^{2}}{{5}^{2}}-\frac{{\left(y - 4\right)}^{2}}{{2}^{2}}=1$; vertices: $\left(2,4\right),\left(-8,4\right)$; foci: $\left(-3+\sqrt{29},4\right),\left(-3-\sqrt{29},4\right)$; asymptotes: $y=\frac{2}{5}\left(x+3\right)+4,y=-\frac{2}{5}\left(x+3\right)+4$

27. $y=\frac{2}{5}\left(x - 3\right)-4,y=-\frac{2}{5}\left(x - 3\right)-4$

29. $y=\frac{3}{4}\left(x - 1\right)+1,y=-\frac{3}{4}\left(x - 1\right)+1$

31.

33.

35.

37.

39.

41.

43.

45. $\frac{{x}^{2}}{9}-\frac{{y}^{2}}{16}=1$

47. $\frac{{\left(x - 6\right)}^{2}}{25}-\frac{{\left(y - 1\right)}^{2}}{11}=1$

49. $\frac{{\left(x - 4\right)}^{2}}{25}-\frac{{\left(y - 2\right)}^{2}}{1}=1$

51. $\frac{{y}^{2}}{16}-\frac{{x}^{2}}{25}=1$

53. $\frac{{y}^{2}}{9}-\frac{{\left(x+1\right)}^{2}}{9}=1$

55. $\frac{{\left(x+3\right)}^{2}}{25}-\frac{{\left(y+3\right)}^{2}}{25}=1$

57. $y\left(x\right)=3\sqrt{{x}^{2}+1},y\left(x\right)=-3\sqrt{{x}^{2}+1}$

59. $y\left(x\right)=1+2\sqrt{{x}^{2}+4x+5},y\left(x\right)=1 - 2\sqrt{{x}^{2}+4x+5}$

61. $\frac{{x}^{2}}{25}-\frac{{y}^{2}}{25}=1$

63. $\frac{{x}^{2}}{100}-\frac{{y}^{2}}{25}=1$

65. $\frac{{x}^{2}}{400}-\frac{{y}^{2}}{225}=1$

67. $\frac{{\left(x - 1\right)}^{2}}{0.25}-\frac{{y}^{2}}{0.75}=1$

69. $\frac{{\left(x - 3\right)}^{2}}{4}-\frac{{y}^{2}}{5}=1$

Section 7.3 Solutions

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 $y=4\left(1\right){x}^{2}$

9. yes ${\left(y - 3\right)}^{2}=4\left(2\right)\left(x - 2\right)$

11. ${y}^{2}=\frac{1}{8}x,V:\left(0,0\right);F:\left(\frac{1}{32},0\right);d:x=-\frac{1}{32}$

13. ${x}^{2}=-\frac{1}{4}y,V:\left(0,0\right);F:\left(0,-\frac{1}{16}\right);d:y=\frac{1}{16}$

15. ${y}^{2}=\frac{1}{36}x,V:\left(0,0\right);F:\left(\frac{1}{144},0\right);d:x=-\frac{1}{144}$

17. ${\left(x - 1\right)}^{2}=4\left(y - 1\right),V:\left(1,1\right);F:\left(1,2\right);d:y=0$

19. ${\left(y - 4\right)}^{2}=2\left(x+3\right),V:\left(-3,4\right);F:\left(-\frac{5}{2},4\right);d:x=-\frac{7}{2}$

21. ${\left(x+4\right)}^{2}=24\left(y+1\right),V:\left(-4,-1\right);F:\left(-4,5\right);d:y=-7$

23. ${\left(y - 3\right)}^{2}=-12\left(x+1\right),V:\left(-1,3\right);F:\left(-4,3\right);d:x=2$

25. ${\left(x - 5\right)}^{2}=\frac{4}{5}\left(y+3\right),V:\left(5,-3\right);F:\left(5,-\frac{14}{5}\right);d:y=-\frac{16}{5}$

27. ${\left(x - 2\right)}^{2}=-2\left(y - 5\right),V:\left(2,5\right);F:\left(2,\frac{9}{2}\right);d:y=\frac{11}{2}$

29. ${\left(y - 1\right)}^{2}=\frac{4}{3}\left(x - 5\right),V:\left(5,1\right);F:\left(\frac{16}{3},1\right);d:x=\frac{14}{3}$

31.

33.

35.

37.

39.

41.

43.

45. ${x}^{2}=-16y$

47. ${\left(y - 2\right)}^{2}=4\sqrt{2}\left(x - 2\right)$

49. ${\left(y+\sqrt{3}\right)}^{2}=-4\sqrt{2}\left(x-\sqrt{2}\right)$

51. ${x}^{2}=y$

53. ${\left(y - 2\right)}^{2}=\frac{1}{4}\left(x+2\right)$

55. ${\left(y-\sqrt{3}\right)}^{2}=4\sqrt{5}\left(x+\sqrt{2}\right)$

57. ${y}^{2}=-8x$

59. ${\left(y+1\right)}^{2}=12\left(x+3\right)$

61. $\left(0,1\right)$

63. At the point 2.25 feet above the vertex.

65. 0.5625 feet

67. ${x}^{2}=-125\left(y - 20\right)$, height is 7.2 feet

69. 2304 feet

Section 7.4 Solutions

1. plotting points with the orientation arrow and a graphing calculator

3. The arrows show the orientation, the direction of motion according to increasing values of $t$.

5. The parametric equations show the different vertical and horizontal motions over time.

7.

9.

11.

13.

15.

17.

19.

21.

23.

25.

27.

29.

31.

33.

35.

37.

39. There will be 100 back-and-forth motions.

41. Take the opposite of the $x\left(t\right)$ equation.

43. The parabola opens up.

45. $\begin{cases}x\left(t\right)=5\cos t\\ y\left(t\right)=5\sin t\end{cases}$

47.

49.

51.

53. $a=4,b=3,c=6,d=1$

55. $a=4,b=2,c=3,d=3$

57.

59.

61. The $y$ -intercept changes.

63. $y\left(x\right)=-16{\left(\frac{x}{15}\right)}^{2}+20\left(\frac{x}{15}\right)$

65. $\begin{cases}x\left(t\right)=64t\cos \left(52^\circ \right)\\ y\left(t\right)=-16{t}^{2}+64t\sin \left(52^\circ \right)\end{cases}$

67. approximately 3.2 seconds

69. 1.6 seconds

71.

73.

Section 7.5 Solutions

1. plotting points with the orientation arrow and a graphing calculator

3. The arrows show the orientation, the direction of motion according to increasing values of $t$.

5. The parametric equations show the different vertical and horizontal motions over time.

7.

9.

11.

13.

15.

17.

19.

21.

23.

25.

27.

29.

31.

33.

35.

37.

39. There will be 100 back-and-forth motions.

41. Take the opposite of the $x\left(t\right)$ equation.

43. The parabola opens up.

45. $\begin{cases}x\left(t\right)=5\cos t\\ y\left(t\right)=5\sin t\end{cases}$

47.

49.

51.

53. $a=4,b=3,c=6,d=1$

55. $a=4,b=2,c=3,d=3$

57.

59.

61. The $y$ -intercept changes.

63. $y\left(x\right)=-16{\left(\frac{x}{15}\right)}^{2}+20\left(\frac{x}{15}\right)$

65. $\begin{cases}x\left(t\right)=64t\cos \left(52^\circ \right)\\ y\left(t\right)=-16{t}^{2}+64t\sin \left(52^\circ \right)\end{cases}$

67. approximately 3.2 seconds

69. 1.6 seconds

71.

73.

Section 7.6 Solutions

1. The $xy$ 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 $xy$ term.

7. $AB=0$, parabola

9. $AB=-4<0$, hyperbola 11. $AB=6>0$, ellipse

13. ${B}^{2}-4AC=0$, parabola

15. ${B}^{2}-4AC=0$, parabola

17. ${B}^{2}-4AC=-96<0$, ellipse 19. $7{{x}^{\prime }}^{2}+9{{y}^{\prime }}^{2}-4=0$ 21. $3{{x}^{\prime }}^{2}+2{x}^{\prime }{y}^{\prime }-5{{y}^{\prime }}^{2}+1=0$ 23. $\theta ={60}^{\circ },11{{x}^{\prime }}^{2}-{{y}^{\prime }}^{2}+\sqrt{3}{x}^{\prime }+{y}^{\prime }-4=0$ 25. $\theta ={150}^{\circ },21{{x}^{\prime }}^{2}+9{{y}^{\prime }}^{2}+4{x}^{\prime }-4\sqrt{3}{y}^{\prime }-6=0$ 27. $\theta \approx {36.9}^{\circ },125{{x}^{\prime }}^{2}+6{x}^{\prime }-42{y}^{\prime }+10=0$ 29. $\theta ={45}^{\circ },3{{x}^{\prime }}^{2}-{{y}^{\prime }}^{2}-\sqrt{2}{x}^{\prime }+\sqrt{2}{y}^{\prime }+1=0$ 31. $\frac{\sqrt{2}}{2}\left({x}^{\prime }+{y}^{\prime }\right)=\frac{1}{2}{\left({x}^{\prime }-{y}^{\prime }\right)}^{2}$

33. $\frac{{\left({x}^{\prime }-{y}^{\prime }\right)}^{2}}{8}+\frac{{\left({x}^{\prime }+{y}^{\prime }\right)}^{2}}{2}=1$

35. $\frac{{\left({x}^{\prime }+{y}^{\prime }\right)}^{2}}{2}-\frac{{\left({x}^{\prime }-{y}^{\prime }\right)}^{2}}{2}=1$

37. $\frac{\sqrt{3}}{2}{x}^{\prime }-\frac{1}{2}{y}^{\prime }={\left(\frac{1}{2}{x}^{\prime }+\frac{\sqrt{3}}{2}{y}^{\prime }-1\right)}^{2}$

39.

41.

43.

45.

47.

49.

51. $\theta ={45}^{\circ }$

53. $\theta ={60}^{\circ }$

55. $\theta \approx {36.9}^{\circ }$

57. [latex]-4\sqrt{6}Section 7.7 Solutions

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 $e=1$ and directrix $\frac{3}{4}$ units below the pole.

9. Hyperbola with $e=2$ and directrix $\frac{5}{2}$ units above the pole.

11. Parabola with $e=1$ and directrix $\frac{3}{10}$ units to the right of the pole.

13. Ellipse with $e=\frac{2}{7}$ and directrix $2$ units to the right of the pole.

15. Hyperbola with $e=\frac{5}{3}$ and directrix $\frac{11}{5}$ units above the pole.

17. Hyperbola with $e=\frac{8}{7}$ and directrix $\frac{7}{8}$ units to the right of the pole.

19. $25{x}^{2}+16{y}^{2}-12y - 4=0$

21. $21{x}^{2}-4{y}^{2}-30x+9=0$

23. $64{y}^{2}=48x+9$

25. $96{y}^{2}-25{x}^{2}+110y+25=0$

27. $3{x}^{2}+4{y}^{2}-2x - 1=0$

29. $5{x}^{2}+9{y}^{2}-24x - 36=0$

31.

33.

35.

37.

39.

41.

43. $r=\frac{4}{5+\cos \theta }$

45. $r=\frac{4}{1+2\sin \theta }$

47. $r=\frac{1}{1+\cos \theta }$

49. $r=\frac{7}{8 - 28\cos \theta }$

51. $r=\frac{12}{2+3\sin \theta }$

53. $r=\frac{15}{4 - 3\cos \theta }$

55. $r=\frac{3}{3 - 3\cos \theta }$

57. $r=\pm \frac{2}{\sqrt{1+\sin \theta \cos \theta }}$

59. $r=\pm \frac{2}{4\cos \theta +3\sin \theta }$