The Ellipse

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.

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

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

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

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

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

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

27. Foci [latex]\left(-3,-1+\sqrt{11}\right),\left(-3,-1-\sqrt{11}\right)[/latex]

29. Focus [latex]\left(0,0\right)[/latex]

31. Foci [latex]\left(-10,30\right),\left(-10,-30\right)[/latex]

33. Center [latex]\left(0,0\right)[/latex], Vertices [latex]\left(4,0\right),\left(-4,0\right),\left(0,3\right),\left(0,-3\right)[/latex], Foci [latex]\left(\sqrt{7},0\right),\left(-\sqrt{7},0\right)[/latex]

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

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

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

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

45. Center [latex]\left(-2,-2\right)[/latex], Vertices [latex]\left(0,-2\right),\left(-4,-2\right),\left(-2,0\right),\left(-2,-4\right)[/latex], Focus [latex]\left(-2,-2\right)[/latex]

47. [latex]\frac{{x}^{2}}{25}+\frac{{y}^{2}}{29}=1[/latex]

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

53. [latex]\frac{{x}^{2}}{81}+\frac{{y}^{2}}{9}=1[/latex]

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

63. [latex]\frac{{x}^{2}}{4{h}^{2}}+\frac{{y}^{2}}{\frac{1}{4}{h}^{2}}=1[/latex]

65. [latex]\frac{{x}^{2}}{400}+\frac{{y}^{2}}{144}=1[/latex]. Distance = 17.32 feet

67. Approximately 51.96 feet

The Hyperbola

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

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

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

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

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

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

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

27. [latex]y=\frac{2}{5}\left(x - 3\right)-4,y=-\frac{2}{5}\left(x - 3\right)-4[/latex]

29. [latex]y=\frac{3}{4}\left(x - 1\right)+1,y=-\frac{3}{4}\left(x - 1\right)+1[/latex]

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45. [latex]\frac{{x}^{2}}{9}-\frac{{y}^{2}}{16}=1[/latex]

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

49. [latex]\frac{{\left(x - 4\right)}^{2}}{25}-\frac{{\left(y - 2\right)}^{2}}{1}=1[/latex]

51. [latex]\frac{{y}^{2}}{16}-\frac{{x}^{2}}{25}=1[/latex]

53. [latex]\frac{{y}^{2}}{9}-\frac{{\left(x+1\right)}^{2}}{9}=1[/latex]

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

61. [latex]\frac{{x}^{2}}{25}-\frac{{y}^{2}}{25}=1[/latex]

63. [latex]\frac{{x}^{2}}{100}-\frac{{y}^{2}}{25}=1[/latex]

65. [latex]\frac{{x}^{2}}{400}-\frac{{y}^{2}}{225}=1[/latex]

67. [latex]\frac{{\left(x - 1\right)}^{2}}{0.25}-\frac{{y}^{2}}{0.75}=1[/latex]

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

The Parabola

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.

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

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

19. [latex]{\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}[/latex]

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

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

25. [latex]{\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}[/latex]

27. [latex]{\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}[/latex]

29. [latex]{\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}[/latex]

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45. [latex]{x}^{2}=-16y[/latex]

47. [latex]{\left(y - 2\right)}^{2}=4\sqrt{2}\left(x - 2\right)[/latex]

49. [latex]{\left(y+\sqrt{3}\right)}^{2}=-4\sqrt{2}\left(x-\sqrt{2}\right)[/latex]

51. [latex]{x}^{2}=y[/latex]

53. [latex]{\left(y - 2\right)}^{2}=\frac{1}{4}\left(x+2\right)[/latex]

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

61. [latex]\left(0,1\right)[/latex]

63. At the point 2.25 feet above the vertex.

65. 0.5625 feet

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

69. 2304 feet

Rotation of Axes

1. The [latex]xy[/latex] 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 [latex]xy[/latex] term.

7. [latex]AB=0[/latex], parabola

9. [latex]AB=-4<0[/latex], hyperbola 11. [latex]AB=6>0[/latex], ellipse

13. [latex]{B}^{2}-4AC=0[/latex], parabola

15. [latex]{B}^{2}-4AC=0[/latex], parabola

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

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

35. [latex]\frac{{\left({x}^{\prime }+{y}^{\prime }\right)}^{2}}{2}-\frac{{\left({x}^{\prime }-{y}^{\prime }\right)}^{2}}{2}=1[/latex]

37. [latex]\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}[/latex]

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51. [latex]\theta ={45}^{\circ }[/latex]

53. [latex]\theta ={60}^{\circ }[/latex]

55. [latex]\theta \approx {36.9}^{\circ }[/latex]

57. [latex]-4\sqrt{6}Conic Sections in Polar Coordinates

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.

7. Parabola with [latex]e=1[/latex] and directrix [latex]\frac{3}{4}[/latex] units below the pole.

9. Hyperbola with [latex]e=2[/latex] and directrix [latex]\frac{5}{2}[/latex] units above the pole.

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

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

15. Hyperbola with [latex]e=\frac{5}{3}[/latex] and directrix [latex]\frac{11}{5}[/latex] units above the pole.

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

19. [latex]25{x}^{2}+16{y}^{2}-12y - 4=0[/latex]

21. [latex]21{x}^{2}-4{y}^{2}-30x+9=0[/latex]

23. [latex]64{y}^{2}=48x+9[/latex]

25. [latex]96{y}^{2}-25{x}^{2}+110y+25=0[/latex]

27. [latex]3{x}^{2}+4{y}^{2}-2x - 1=0[/latex]

29. [latex]5{x}^{2}+9{y}^{2}-24x - 36=0[/latex]

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43. [latex]r=\frac{4}{5+\cos \theta }[/latex]

45. [latex]r=\frac{4}{1+2\sin \theta }[/latex]

47. [latex]r=\frac{1}{1+\cos \theta }[/latex]

49. [latex]r=\frac{7}{8 - 28\cos \theta }[/latex]

51. [latex]r=\frac{12}{2+3\sin \theta }[/latex]

53. [latex]r=\frac{15}{4 - 3\cos \theta }[/latex]

55. [latex]r=\frac{3}{3 - 3\cos \theta }[/latex]

57. [latex]r=\pm \frac{2}{\sqrt{1+\sin \theta \cos \theta }}[/latex]

59. [latex]r=\pm \frac{2}{4\cos \theta +3\sin \theta }[/latex]