Conic Sections: Get Stronger Answer Key

Circles

  1. [latex]x^2 + y^2 = 49[/latex]
  2. [latex]x^2 + y^2 = 2[/latex]
  3. [latex](x - 3)^2 + (y - 5)^2 = 1[/latex]
  4. [latex](x - 1.5)^2 + (y + 3.5)^2 = 6.25[/latex]
  5. [latex](x - 3)^2 + (y + 2)^2 = 64[/latex]
  6. [latex](x - 4)^2 + (y - 4)^2 = 8[/latex]
    1. The circle is centered at [latex](-5, -3)[/latex] with a radius of [latex]1[/latex].
    2. This graph shows a circle with center at (negative 5, negative 3) and a radius of 1.
    1. The circle is centered at [latex](4, -2)[/latex] with a radius of [latex]4[/latex].
    2. This graph shows circle with center at (4, negative 2) and a radius of 4.
    1. The circle is centered at [latex](0, -2)[/latex] with a radius of [latex]5[/latex].
    2. This graph shows circle with center at (negative 2, 5) and a radius of 5.
    1. The circle is centered at [latex](1.5, -2.5)[/latex] with a radius of [latex]0.5[/latex].
    2. This graph shows circle with center at (1.5, 2.5) and a radius of 0.5
    1. The circle is centered at [latex](0, 0)[/latex] with a radius of [latex]8[/latex].
    2. This graph shows circle with center at (0, 0) and a radius of 8.
    1. The circle is centered at [latex](0, 0)[/latex] with a radius of [latex]2[/latex].
    2. This graph shows circle with center at (0, 0) and a radius of 2.
    1. Center: [latex](-1, -3)[/latex], radius: [latex]1[/latex]
    2. This graph shows circle with center at (negative 1, negative 3) and a radius of 1.
    1. Center: [latex](2, -5)[/latex], radius: [latex]6[/latex]
    2. This graph shows circle with center at (2, negative 5) and a radius of 6.
    1. Center: [latex](0, -3)[/latex], radius: [latex]2[/latex]
    2. This graph shows circle with center at (0, negative 3) and a radius of 2.
    1. Center: [latex](-2, 0)[/latex], radius: [latex]2[/latex]
    2. This graph shows circle with center at (negative 2, 0) and a radius of 2.

Ellipses

  1. yes; [latex]\dfrac{x^2}{3^2} + \dfrac{y^2}{2^2} = 1[/latex]
  2. yes; [latex]\dfrac{x^2}{(\dfrac{1}{2})^2} + \dfrac{y^2}{(\dfrac{1}{3})^2} = 1[/latex]
  3. [latex]\dfrac{x^2}{2^2} + \dfrac{y^2}{7^2} = 1[/latex]; Endpoints of major axis [latex](0, 7)[/latex] and [latex](0, -7)[/latex]. Endpoints of minor axis [latex](2, 0)[/latex] and [latex](-2, 0)[/latex]. Foci at [latex](0, 3\sqrt{5})[/latex], [latex](0, -3\sqrt{5})[/latex].
  4. [latex]\dfrac{x^2}{(1)^2} + \dfrac{y^2}{(\dfrac{1}{3})^2} = 1[/latex]; Endpoints of major axis [latex](1, 0)[/latex] and [latex](-1, 0)[/latex]. Endpoints of minor axis [latex](0, \dfrac{1}{3})[/latex], [latex](0, -\dfrac{1}{3})[/latex]. Foci at [latex](\dfrac{2\sqrt{2}}{3}, 0)[/latex], [latex](-\dfrac{2\sqrt{2}}{3}, 0)[/latex].
  5. [latex]\dfrac{(x-2)^2}{7^2} + \dfrac{(y-4)^2}{5^2} = 1[/latex]; Endpoints of major axis [latex](9, 4)[/latex], [latex](-5, 4)[/latex]. Endpoints of minor axis [latex](2, 9)[/latex], [latex](2, -1)[/latex]. Foci at [latex](2 + 2\sqrt{6}, 4)[/latex], [latex](2 - 2\sqrt{6}, 4)[/latex].
  6. [latex]\dfrac{(x+5)^2}{2^2} + \dfrac{(y-7)^2}{3^2} = 1[/latex]; Endpoints of major axis [latex](-5, 10)[/latex], [latex](-5, 4)[/latex]. Endpoints of minor axis [latex](-3, 7)[/latex], [latex](-7, 7)[/latex]. Foci at [latex](-5, 7 + \sqrt{5})[/latex], [latex](-5, 7 - \sqrt{5})[/latex].
  7. [latex]\dfrac{(x-1)^2}{3^2} + \dfrac{(y-4)^2}{2^2} = 1[/latex]; Endpoints of major axis [latex](4, 4)[/latex], [latex](-2, 4)[/latex]. Endpoints of minor axis [latex](1, 6)[/latex], [latex](1, 2)[/latex]. Foci at [latex](1 + \sqrt{5}, 4)[/latex], [latex](1 - \sqrt{5}, 4)[/latex].
  8. [latex]\dfrac{(x-3)^2}{(3\sqrt{2})^2} + \dfrac{(y-5)^2}{(\sqrt{2})^2} = 1[/latex]; Endpoints of major axis [latex](3 + 3\sqrt{2}, 5)[/latex], [latex](3 - 3\sqrt{2}, 5)[/latex]. Endpoints of minor axis [latex](3, 5 + \sqrt{2})[/latex], [latex](3, 5 - \sqrt{2})[/latex]. Foci at [latex](7, 5)[/latex], [latex](-1, 5)[/latex].
  9. [latex]\dfrac{(x+5)^2}{(5)^2} + \dfrac{(y-2)^2}{(2)^2} = 1[/latex]; Endpoints of major axis [latex](0, 2)[/latex], [latex](-10, 2)[/latex]. Endpoints of minor axis [latex](-5, 4)[/latex], [latex](-5, 0)[/latex]. Foci at [latex](-5 + \sqrt{21}, 2)[/latex], [latex](-5 - \sqrt{21}, 2)[/latex].
  10. [latex]\dfrac{(x+3)^2}{(5)^2} + \dfrac{(y+4)^2}{(2)^2} = 1[/latex]; Endpoints of major axis [latex](2, -4)[/latex], [latex](-8, -4)[/latex]. Endpoints of minor axis [latex](-3, -2)[/latex], [latex](-3, -6)[/latex]. Foci at [latex](-3 + \sqrt{21}, -4)[/latex], [latex](-3 - \sqrt{21}, -4)[/latex].
  11. Foci [latex](-3, -1 + \sqrt{11})[/latex], [latex](-3, -1 - \sqrt{11})[/latex]
  12. Focus [latex](0, 0)[/latex]
  13. Foci [latex](-10, 30)[/latex], [latex](-10, -30)[/latex]
  14. Center [latex](0, 0)[/latex], Vertices [latex](4, 0)[/latex], [latex](-4, 0)[/latex], [latex](0, 3)[/latex], [latex](0, -3)[/latex], Foci [latex](\sqrt{7}, 0)[/latex], [latex](-\sqrt{7}, 0)[/latex]
    Graph of an ellipse centered at the origin, extending from -4 to 4 on the x-axis and -3 to 3 on the y-axis, with grid lines and labeled axes.
  15. Center [latex](0, 0)[/latex], Vertices [latex](\dfrac{1}{9}, 0)[/latex], [latex](-\dfrac{1}{9}, 0)[/latex], [latex](0, \dfrac{1}{7})[/latex], [latex](0, -\dfrac{1}{7})[/latex], Foci [latex](0, \dfrac{4\sqrt{2}}{63})[/latex], [latex](0, -\dfrac{4\sqrt{2}}{63})[/latex]
    Graph of a narrow ellipse centered at the origin, extending from -0.2 to 0.2 on the x-axis and from -0.2 to 0.2 on the y-axis, with grid lines and labeled axes.
  16. Center [latex](-3, 3)[/latex], Vertices [latex](0, 3)[/latex], [latex](-6, 3)[/latex], [latex](-3, 0)[/latex], [latex](-3, 6)[/latex], Focus [latex](-3, 3)[/latex]
    Note that this ellipse is a circle. The circle has only one focus, which coincides with the center.
    Graph of a circle centered slightly left of the y-axis, extending from -7.5 to -2.5 on the x-axis and from 0 to 5 on the y-axis, with grid lines and labeled axes.
  17. Center [latex](1, 1)[/latex], Vertices [latex](5, 1)[/latex], [latex](-3, 1)[/latex], [latex](1, 3)[/latex], [latex](1, -1)[/latex], Foci [latex](1 + 2\sqrt{3}, 1)[/latex], [latex](1 - 2\sqrt{3}, 1)[/latex]
    Graph of an ellipse centered at the origin, extending from -5 to 5 on the x-axis and from -3 to 3 on the y-axis, with grid lines and labeled axes.
  18. Center [latex](-4, 5)[/latex], Vertices [latex](-2, 5)[/latex], [latex](-6, 4)[/latex], [latex](-4, 6)[/latex], [latex](-4, 4)[/latex], Foci [latex](-4 + \sqrt{3}, 5)[/latex], [latex](-4 - \sqrt{3}, 5)[/latex]
    Graph of a small ellipse centered above the x-axis, extending from -1 to 1 on the x-axis and from 4.5 to 5.5 on the y-axis, with grid lines and labeled axes.
  19. Center [latex](-2, 1)[/latex], Vertices [latex](0, 1)[/latex], [latex](-4, 1)[/latex], [latex](-2, 5)[/latex], [latex](-2, -3)[/latex], Foci [latex](-2, 1 + 2\sqrt{3})[/latex], [latex](-2, 1 - 2\sqrt{3})[/latex]
    Graph of a tall, narrow ellipse centered at the origin, extending from -2.5 to 2.5 on the y-axis and from -1 to 1 on the x-axis, with grid lines and labeled axes.
  20. Center [latex](-2, -2)[/latex], Vertices [latex](0, -2)[/latex], [latex](-4, -2)[/latex], [latex](-2, 0)[/latex], [latex](-2, -4)[/latex], Focus [latex](-2, -2)[/latex]
    Graph of a circle centered to the left of the y-axis, extending from -5 to -1 on the x-axis and from -5 to 0 on the y-axis, with grid lines and labeled axes.
  21. [latex]\dfrac{x^2}{25} + \dfrac{y^2}{29} = 1[/latex]
  22. [latex]\dfrac{(x-4)^2}{25} + \dfrac{(y-2)^2}{1} = 1[/latex]
  23. [latex]\dfrac{(x+3)^2}{16} + \dfrac{(y-4)^2}{4} = 1[/latex]
  24. [latex]\dfrac{x^2}{81} + \dfrac{y^2}{9} = 1[/latex]
  25. [latex]\dfrac{(x+2)^2}{4} + \dfrac{(y-2)^2}{9} = 1[/latex]
  26. [latex]\dfrac{x^2}{4h^2} + \dfrac{y^2}{1h^2} = 1[/latex]
  27. [latex]\dfrac{x^2}{400} + \dfrac{y^2}{144} = 1[/latex]. Distance = [latex]17.32[/latex] feet
  28. Approximately [latex]51.96[/latex] feet

Hyperbolas

  1. yes; [latex]\dfrac{x^2}{6^2} - \dfrac{y^2}{3^2} = 1[/latex]
  2. yes; [latex]\dfrac{x^2}{4^2} - \dfrac{y^2}{5^2} = 1[/latex]
  3. [latex]\dfrac{x^2}{5^2} - \dfrac{y^2}{6^2} = 1[/latex]; vertices: [latex](5, 0)[/latex], [latex](-5, 0)[/latex]; foci: [latex](\sqrt{61}, 0)[/latex], [latex](-\sqrt{61}, 0)[/latex]; asymptotes: [latex]y = \dfrac{6}{5}x[/latex], [latex]y = -\dfrac{6}{5}x[/latex]
  4. [latex]\dfrac{y^2}{2^2} - \dfrac{x^2}{9^2} = 1[/latex]; vertices: [latex](0, 2)[/latex], [latex](0, -2)[/latex]; foci: [latex](0, \sqrt{85})[/latex], [latex](0, -\sqrt{85})[/latex]; asymptotes: [latex]y = \dfrac{2}{9}x[/latex], [latex]y = -\dfrac{2}{9}x[/latex]
  5. [latex]\dfrac{(x-1)^2}{3^2} - \dfrac{(y-2)^2}{4^2} = 1[/latex]; vertices: [latex](4, 2)[/latex], [latex](-2, 2)[/latex]; foci: [latex](6, 2)[/latex], [latex](-4, 2)[/latex]; asymptotes: [latex]y = \dfrac{4}{3}(x - 1) + 2[/latex], [latex]y = -\dfrac{4}{3}(x - 1) + 2[/latex]
  6. [latex]\dfrac{(x-2)^2}{7^2} - \dfrac{(y+7)^2}{7^2} = 1[/latex]; vertices: [latex](9, -7)[/latex], [latex](-5, -7)[/latex]; foci: [latex](2 + 7\sqrt{2}, -7)[/latex], [latex](2 - 7\sqrt{2}, -7)[/latex]; asymptotes: [latex]y = x - 9[/latex], [latex]y = -x - 5[/latex]
  7. [latex]\dfrac{(x+3)^2}{2^2} - \dfrac{(y-3)^2}{3^2} = 1[/latex]; vertices: [latex](0, 3)[/latex], [latex](-6, 3)[/latex]; foci: [latex](-3 + 3\sqrt{2}, 1)[/latex], [latex](-3 - 3\sqrt{2}, 1)[/latex]; asymptotes: [latex]y = x + 6[/latex], [latex]y = -x[/latex]
  8. [latex]\dfrac{(y-4)^2}{2^2} - \dfrac{(x-3)^2}{4^2} = 1[/latex]; vertices: [latex](3, 6)[/latex], [latex](3, 2)[/latex]; foci: [latex](3, 4 + 2\sqrt{5})[/latex], [latex](3, 4 - 2\sqrt{5})[/latex]; asymptotes: [latex]y = \dfrac{1}{2}(x - 3) + 4[/latex], [latex]y = -\dfrac{1}{2}(x - 3) + 4[/latex]
  9. [latex]\dfrac{(y+5)^2}{7^2} - \dfrac{(x+1)^2}{70^2} = 1[/latex]; vertices: [latex](-1, 2)[/latex], [latex](-1, -12)[/latex]; foci: [latex](-1, -5 + 7\sqrt{101})[/latex], [latex](-1, -5 - 7\sqrt{101})[/latex]; asymptotes: [latex]y = \dfrac{1}{10}(x + 1) - 5[/latex], [latex]y = -\frac{1}{10}(x + 1) - 5[/latex]
  10. [latex]\dfrac{(x+3)^2}{5^2} - \dfrac{(y-4)^2}{2^2} = 1[/latex]; vertices: [latex](2, 4)[/latex], [latex](-8, 4)[/latex]; foci: [latex](-3 + \sqrt{29}, 4)[/latex], [latex](-3 - \sqrt{29}, 4)[/latex]; asymptotes: [latex]y = \dfrac{2}{5}(x + 3) + 4[/latex], [latex]y = -\dfrac{2}{5}(x + 3) + 4[/latex]
  11. [latex]y = \dfrac{2}{5}(x - 3) - 4[/latex], [latex]y = -\dfrac{2}{5}(x - 3) - 4[/latex]
  12. [latex]y = \dfrac{3}{4}(x - 1) + 1[/latex], [latex]y = -\dfrac{3}{4}(x - 1) + 1[/latex]
  13. Graph of a hyperbola with vertices at (-7, 0) and (7, 0) and foci at approximately (-8.06, 0) and (8.06, 0). The hyperbola opens horizontally along the x-axis, with grid lines, labeled axes, and arrows indicating the curve's direction extending outward.
  14. Graph of a hyperbola opening vertically along the y-axis with vertices at (0, 3) and (0, -3) and foci at approximately (0, 5.83) and (0, -5.83). The graph includes grid lines, labeled axes, and arrows indicating the curve extending upward and downward.
  15. Graph of a hyperbola opening vertically along the y-axis with vertices at (4, -2) and (4, -8) and foci at approximately (4, 0.83) and (4, -10.83). The graph includes grid lines, labeled axes, and arrows indicating the curve extending upward and downward.
  16. Graph of a hyperbola opening vertically along the y-axis with vertices at (3, 0) and (3, 6) and foci at approximately (3, 7.24) and (3, -1.24). The graph includes grid lines, labeled axes, and arrows indicating the curve extending upward and downward.
  17. Graph of a hyperbola opening horizontally along the x-axis with vertices at (-1, -2) and (9, -2) and foci at approximately (-1.1, -2) and (9.1, -2). The graph includes grid lines, labeled axes, and arrows indicating the curve extending outward to the left and right.
  18. Graph of a hyperbola opening vertically along the y-axis with vertices at (-4, -4) and (2, -4) and foci at approximately (-9.54, -4) and (7.54, -4). The graph includes grid lines, labeled axes, and arrows indicating the curves extending upward and downward.
  19. Graph of a hyperbola opening vertically along the y-axis with vertices at (5, 15) and (5, -5) and foci at approximately (5, 15.05) and (5, -5.05). The graph includes grid lines, labeled axes, and arrows indicating the curves extending upward and downward.
  20. [latex]\dfrac{x^2}{9} - \dfrac{y^2}{16} = 1[/latex]
  21. [latex]\dfrac{(x-6)^2}{25} - \dfrac{(y-1)^2}{11} = 1[/latex]
  22. [latex]\dfrac{(x-4)^2}{25} - \dfrac{(y-2)^2}{1} = 1[/latex]
  23. [latex]\dfrac{x^2}{16} - \dfrac{y^2}{25} = 1[/latex]
  24. [latex]\dfrac{y^2}{9} - \dfrac{(x+1)^2}{9} = 1[/latex]
  25. [latex]\dfrac{(x+3)^2}{25} - \dfrac{(y+3)^2}{25} = 1[/latex]
  26. [latex]\dfrac{x^2}{25} - \dfrac{y^2}{25} = 1[/latex]
    Graph of a hyperbola opening horizontally along the x-axis with the origin labeled as "Fountain." The curves extend outward to the left and right, with grid lines, labeled axes, and arrows indicating the direction of the curves.
  27. [latex]\dfrac{x^2}{100} - \dfrac{y^2}{25} = 1[/latex]
    Graph of a hyperbola opening horizontally along the x-axis with the origin labeled as "Fountain." The curves extend outward to the left and right, with grid lines, labeled axes, and arrows showing the direction of the curves.
  28. [latex]\dfrac{x^2}{400} - \dfrac{y^2}{225} = 1[/latex]
    Graph of a hyperbola opening horizontally along the x-axis with the origin labeled as "Fountain." The curves extend outward to the left and right, spanning from -40 to 40 on the x-axis and from -24 to 24 on the y-axis, with grid lines, labeled axes, and arrows showing the curve's direction.
  29. [latex]4(x-1)^2 - y2^2 = 16[/latex]
  30. [latex]\dfrac{(x-h)^2}{a^2} - \dfrac{(y-k)^2}{b^2} = (x-3)^2 - 9y^2 = 4[/latex]

Parabolas

  1. yes [latex]x^2 = 4(\dfrac{1}{16})y[/latex]
  2. yes [latex](y-3)^2 = 4(2)(x-2)[/latex]
  3. [latex]y^2 = \dfrac{1}{8}x[/latex], V: [latex](0,0)[/latex]; F: [latex](\dfrac{1}{32},0)[/latex]; d: [latex]x = -\dfrac{1}{32}[/latex]
  4. [latex]x^2 = -\dfrac{1}{4}y[/latex], V: [latex](0,0)[/latex]; F: [latex](0,-\dfrac{1}{16})[/latex]; d: [latex]y = \dfrac{1}{16}[/latex]
  5. [latex]y^2 = \dfrac{1}{36}x[/latex], V: [latex](0,0)[/latex]; F: [latex](\dfrac{1}{144},0)[/latex]; d: [latex]x = -\dfrac{1}{144}[/latex]
  6. [latex](x-1)^2 = 4(y-1)[/latex], V: [latex](1,1)[/latex]; F: [latex](1,2)[/latex]; d: [latex]y = 0[/latex]
  7. [latex](y-4)^2 = 2(x+3)[/latex], V: [latex](-3,4)[/latex]; F: [latex](-\dfrac{5}{2},4)[/latex]; d: [latex]x = -\dfrac{7}{2}[/latex]
  8. [latex](x+4)^2 = 24(y+1)[/latex], V: [latex](-4,-1)[/latex]; F: [latex](-4,5)[/latex]; d: [latex]y = -7[/latex]
  9. [latex](y-3)^2 = -12(x+1)[/latex], V: [latex](-1,3)[/latex]; F: [latex](-4,3)[/latex]; d: [latex]x = 2[/latex]
  10. [latex](x-5)^2 = \dfrac{8}{5}(y+3)[/latex], V: [latex](5,-3)[/latex]; F: [latex](5,-\dfrac{14}{5})[/latex]; d: [latex]y = -\dfrac{16}{5}[/latex]
  11. [latex](x-2)^2 = -2(y-5)[/latex], V: [latex](2,5)[/latex]; F: [latex](2,\dfrac{9}{2})[/latex]; d: [latex]y = \dfrac{11}{2}[/latex]
  12. [latex](y-1)^2 = \dfrac{4}{3}(x-5)[/latex], V: [latex](5,1)[/latex]; F: [latex](\dfrac{16}{3},1)[/latex]; d: [latex]x = \dfrac{14}{3}[/latex]
  13. Graph of a parabola opening to the right along the x-axis with a focus at (2, 0) and a vertical line labeled x=−2 as the directrix. The graph includes grid lines, labeled axes, and an arrow indicating the curve's direction.
  14. Graph of a parabola opening upward along the y-axis with a focus at (0, 9) and a horizontal line labeled y=−9 as the directrix. The graph includes grid lines, labeled axes, and arrows indicating the curve’s direction.
  15. Graph of a parabola opening to the left along the x-axis with a focus at (7/3, 2) and a vertical line labeled x = -5/3 as the directrix. The graph includes grid lines, labeled axes, and arrows indicating the curve’s direction.
  16. Graph of a parabola opening to the left along the x-axis with a focus at (23/6, -5) and a vertical line labeled x = 25/6 as the directrix. The graph includes grid lines, labeled axes, and arrows indicating the curve’s direction.
  17. Graph of a downward-opening parabola with a focus at (-4, -2) and a horizontal line labeled y = 0 as the directrix. The graph includes grid lines, labeled axes, and arrows indicating the curve’s downward direction.
  18. Graph of a parabola opening to the right along the x-axis with a focus at (0, -5) and a vertical line labeled x = -4 as the directrix. The graph includes grid lines, labeled axes, and arrows indicating the curve’s direction.
  19. Graph of a parabola opening to the right along the x-axis with a focus at (8, -1) and a vertical line labeled x = 2 as the directrix. The graph includes grid lines, labeled axes, and arrows indicating the curve’s direction.
  20. [latex]x^2 = -16y[/latex]
  21. [latex](y-2)^2 = 4\sqrt{2}(x-2)[/latex]
  22. [latex](y+\sqrt{3})^2 = -4\sqrt{2}(x-\sqrt{2})[/latex]
  23. [latex]x^2 = y[/latex]
  24. [latex](y-2)^2 = \dfrac{1}{4}(x+2)[/latex]
  25. [latex](y-\sqrt{3})^2 = 4\sqrt{5}(x+\sqrt{2})[/latex]
  26. [latex](0,1)[/latex]
  27. At the point [latex]2.25[/latex] feet above the vertex.
  28. [latex]0.5625[/latex] feet
  29. [latex]x^2 = -125(y-20)[/latex], height is [latex]7.2[/latex] feet
  30. [latex]2304[/latex] feet