{"id":2373,"date":"2025-08-13T00:59:14","date_gmt":"2025-08-13T00:59:14","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2373"},"modified":"2026-02-18T20:30:11","modified_gmt":"2026-02-18T20:30:11","slug":"conics-get-stronger","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/conics-get-stronger\/","title":{"raw":"Conics: Get Stronger","rendered":"Conics: Get Stronger"},"content":{"raw":"

The Ellipse<\/h1>\r\n1. Define an ellipse in terms of its foci.\r\n\r\n3. What special case of the ellipse do we have when the major and minor axis are of the same length?\r\n\r\n5. What can be said about the symmetry of the graph of an ellipse with center at the origin and foci along the y-axis?\r\n\r\nFor the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.\r\n\r\n11. [latex]\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{49}=1[\/latex]\r\n\r\n13. [latex]{x}^{2}+9{y}^{2}=1[\/latex]\r\n\r\n15. [latex]\\frac{{\\left(x - 2\\right)}^{2}}{49}+\\frac{{\\left(y - 4\\right)}^{2}}{25}=1[\/latex]\r\n\r\n17. [latex]\\frac{{\\left(x+5\\right)}^{2}}{4}+\\frac{{\\left(y - 7\\right)}^{2}}{9}=1[\/latex]\r\n\r\n19. [latex]4{x}^{2}-8x+9{y}^{2}-72y+112=0[\/latex]\r\n\r\n23. [latex]4{x}^{2}+40x+25{y}^{2}-100y+100=0[\/latex]\r\n\r\nFor the following exercises, find the foci for the given ellipses.\r\n\r\n27. [latex]\\frac{{\\left(x+3\\right)}^{2}}{25}+\\frac{{\\left(y+1\\right)}^{2}}{36}=1[\/latex]\r\n\r\n29. [latex]{x}^{2}+{y}^{2}=1[\/latex]\r\n\r\n31. [latex]10{x}^{2}+{y}^{2}+200x=0[\/latex]\r\n\r\nFor the following exercises, graph the given ellipses, noting center, vertices, and foci.\r\n\r\n33. [latex]\\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{9}=1[\/latex]\r\n\r\n35. [latex]81{x}^{2}+49{y}^{2}=1[\/latex]\r\n\r\n37. [latex]\\frac{{\\left(x+3\\right)}^{2}}{9}+\\frac{{\\left(y - 3\\right)}^{2}}{9}=1[\/latex]\r\n\r\n39. [latex]4{x}^{2}-8x+16{y}^{2}-32y - 44=0[\/latex]\r\n\r\n41. [latex]{x}^{2}+8x+4{y}^{2}-40y+112=0[\/latex]\r\n\r\n45. [latex]4{x}^{2}+16x+4{y}^{2}+16y+16=0[\/latex]\r\n\r\nFor the following exercises, use the given information about the graph of each ellipse to determine its equation.\r\n\r\n47. Center at the origin, symmetric with respect to the x<\/em>- and y<\/em>-axes, focus at [latex]\\left(0,-2\\right)[\/latex], and point on graph [latex]\\left(5,0\\right)[\/latex].\r\n\r\n49. Center [latex]\\left(4,2\\right)[\/latex] ; vertex [latex]\\left(9,2\\right)[\/latex] ; one focus: [latex]\\left(4+2\\sqrt{6},2\\right)[\/latex] .\r\n\r\nFor the following exercises, given the graph of the ellipse, determine its equation.\r\n\r\n53.\r\n\"\"\r\n\r\n55.\r\n\"\"\r\n\r\n63. Find the equation of the ellipse that will just fit inside a box that is four times as wide as it is high. Express in terms of [latex]h[\/latex], the height.\r\n\r\n65. An arch has the shape of a semi-ellipse. The arch has a height of 12 feet and a span of 40 feet. Find an equation for the ellipse, and use that to find the distance from the center to a point at which the height is 6 feet. Round to the nearest hundredth.\r\n\r\n67. A person in a whispering gallery standing at one focus of the ellipse can whisper and be heard by a person standing at the other focus because all the sound waves that reach the ceiling are reflected to the other person. If a whispering gallery has a length of 120 feet, and the foci are located 30 feet from the center, find the height of the ceiling at the center.\r\n

The Hyperbola<\/h1>\r\n5. Where must the center of hyperbola be relative to its foci?\r\n\r\nFor the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.\r\n\r\n11. [latex]\\frac{{x}^{2}}{25}-\\frac{{y}^{2}}{36}=1[\/latex]\r\n\r\n13. [latex]\\frac{{y}^{2}}{4}-\\frac{{x}^{2}}{81}=1[\/latex]\r\n\r\n15. [latex]\\frac{{\\left(x - 1\\right)}^{2}}{9}-\\frac{{\\left(y - 2\\right)}^{2}}{16}=1[\/latex]\r\n\r\n17. [latex]\\frac{{\\left(x - 2\\right)}^{2}}{49}-\\frac{{\\left(y+7\\right)}^{2}}{49}=1[\/latex]\r\n\r\n19. [latex]-9{x}^{2}-54x+9{y}^{2}-54y+81=0[\/latex]\r\n\r\n23. [latex]{x}^{2}+2x - 100{y}^{2}-1000y+2401=0[\/latex]\r\n\r\nFor the following exercises, find the equations of the asymptotes for each hyperbola.\r\n\r\n27.\u00a0[latex]\\frac{{\\left(x - 3\\right)}^{2}}{{5}^{2}}-\\frac{{\\left(y+4\\right)}^{2}}{{2}^{2}}=1[\/latex]\r\n\r\n29.\u00a0[latex]9{x}^{2}-18x - 16{y}^{2}+32y - 151=0[\/latex]\r\n\r\nFor the following exercises, sketch a graph of the hyperbola, labeling vertices and foci.\r\n\r\n31. [latex]\\frac{{x}^{2}}{49}-\\frac{{y}^{2}}{16}=1[\/latex]\r\n\r\n33. [latex]\\frac{{y}^{2}}{9}-\\frac{{x}^{2}}{25}=1[\/latex]\r\n\r\n35. [latex]\\frac{{\\left(y+5\\right)}^{2}}{9}-\\frac{{\\left(x - 4\\right)}^{2}}{25}=1[\/latex]\r\n\r\n37. [latex]\\frac{{\\left(y - 3\\right)}^{2}}{9}-\\frac{{\\left(x - 3\\right)}^{2}}{9}=1[\/latex]\r\n\r\n39. [latex]{x}^{2}-8x - 25{y}^{2}-100y - 109=0[\/latex]\r\n\r\n41. [latex]64{x}^{2}+128x - 9{y}^{2}-72y - 656=0[\/latex]\r\n\r\n43. [latex]-100{x}^{2}+1000x+{y}^{2}-10y - 2575=0[\/latex]\r\n\r\nFor the following exercises, given information about the graph of the hyperbola, find its equation.\r\n\r\n45. Vertices at [latex]\\left(3,0\\right)[\/latex] and [latex]\\left(-3,0\\right)[\/latex] and one focus at [latex]\\left(5,0\\right)[\/latex].\r\n\r\n47. Vertices at [latex]\\left(1,1\\right)[\/latex] and [latex]\\left(11,1\\right)[\/latex] and one focus at [latex]\\left(12,1\\right)[\/latex].\r\n\r\n49. Center: [latex]\\left(4,2\\right)[\/latex]; vertex: [latex]\\left(9,2\\right)[\/latex]; one focus: [latex]\\left(4+\\sqrt{26},2\\right)[\/latex].\r\n\r\nFor the following exercises, given the graph of the hyperbola, find its equation.\r\n\r\n51.\r\n\"\"\r\n\r\n53.\r\n\"\"\r\n\r\n55.\r\n\"\"\r\n\r\nFor the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph.\r\n\r\n61. The hedge will follow the asymptotes [latex]y=x\\text{ and }y=-x[\/latex], and its closest distance to the center fountain is 5 yards.\r\n\r\n63. The hedge will follow the asymptotes [latex]y=\\frac{1}{2}x[\/latex] and [latex]y=-\\frac{1}{2}x[\/latex], and its closest distance to the center fountain is 10 yards.\r\n\r\n65. The hedge will follow the asymptotes [latex]\\text{ }y=\\frac{3}{4}x\\text{ and }y=-\\frac{3}{4}x[\/latex], and its closest distance to the center fountain is 20 yards.\r\n\r\nFor the following exercises, assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the object's path. Give the equation of the flight path of each object using the given information.\r\n\r\n67. The object enters along a path approximated by the line [latex]y=2x - 2[\/latex] and passes within 0.5 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line [latex]y=-2x+2[\/latex].\r\n\r\n69. The object enters along a path approximated by the line [latex]y=\\frac{1}{3}x - 1[\/latex] and passes within 1 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line [latex]\\text{ }y=-\\frac{1}{3}x+1[\/latex].\r\n

The Parabola<\/h1>\r\n1. Define a parabola in terms of its focus and directrix.\r\n\r\n3. If the equation of a parabola is written in standard form and [latex]p[\/latex] is negative and the directrix is a horizontal line, then what can we conclude about its graph?\r\n\r\n5. As the graph of a parabola becomes wider, what will happen to the distance between the focus and directrix?\r\n\r\nFor the following exercises, rewrite the given equation in standard form, and then determine the vertex [latex]\\left(V\\right)[\/latex], focus [latex]\\left(F\\right)[\/latex], and directrix [latex]\\text{ }\\left(d\\right)\\text{ }[\/latex] of the parabola.\r\n\r\n11. [latex]x=8{y}^{2}[\/latex]\r\n\r\n13. [latex]y=-4{x}^{2}[\/latex]\r\n\r\n19. [latex]{\\left(y - 4\\right)}^{2}=2\\left(x+3\\right)[\/latex]\r\n\r\n21. [latex]{\\left(x+4\\right)}^{2}=24\\left(y+1\\right)[\/latex]\r\n\r\n23. [latex]{y}^{2}+12x - 6y+21=0[\/latex]\r\n\r\n25. [latex]5{x}^{2}-50x - 4y+113=0[\/latex]\r\n\r\n27. [latex]{x}^{2}-4x+2y - 6=0[\/latex]\r\n\r\n29. [latex]3{y}^{2}-4x - 6y+23=0[\/latex]\r\n\r\nFor the following exercises, graph the parabola, labeling the focus and the directrix.\r\n\r\n31. [latex]x=\\frac{1}{8}{y}^{2}[\/latex]\r\n\r\n33. [latex]y=\\frac{1}{36}{x}^{2}[\/latex]\r\n\r\n35. [latex]{\\left(y - 2\\right)}^{2}=-\\frac{4}{3}\\left(x+2\\right)[\/latex]\r\n\r\n37. [latex]-6{\\left(y+5\\right)}^{2}=4\\left(x - 4\\right)[\/latex]\r\n\r\n39. [latex]{x}^{2}+8x+4y+20=0[\/latex]\r\n\r\n41. [latex]{y}^{2}-8x+10y+9=0[\/latex]\r\n\r\n43. [latex]{y}^{2}+2y - 12x+61=0[\/latex]\r\n\r\nFor the following exercises, find the equation of the parabola given information about its graph.\r\n\r\n45. Vertex is [latex]\\left(0,0\\right)[\/latex]; directrix is [latex]y=4[\/latex], focus is [latex]\\left(0,-4\\right)[\/latex].\r\n\r\n47. Vertex is [latex]\\left(2,2\\right)[\/latex]; directrix is [latex]x=2-\\sqrt{2}[\/latex], focus is [latex]\\left(2+\\sqrt{2},2\\right)[\/latex].\r\n\r\n49. Vertex is [latex]\\left(\\sqrt{2},-\\sqrt{3}\\right)[\/latex]; directrix is [latex]x=2\\sqrt{2}[\/latex], focus is [latex]\\left(0,-\\sqrt{3}\\right)[\/latex].\r\n\r\nFor the following exercises, determine the equation for the parabola from its graph.\r\n\r\n51.\r\n\"\"\r\n\r\n53.\r\n\"\"\r\n\r\n55.\r\n\"\"\r\n\r\n61.\u00a0The mirror in an automobile headlight has a parabolic cross-section with the light bulb at the focus. On a schematic, the equation of the parabola is given as [latex]{x}^{2}=4y[\/latex]. At what coordinates should you place the light bulb?\r\n\r\n63. A satellite dish is shaped like a paraboloid of revolution. This means that it can be formed by rotating a parabola around its axis of symmetry. The receiver is to be located at the focus. If the dish is 12 feet across at its opening and 4 feet deep at its center, where should the receiver be placed?\r\n\r\n65. A searchlight is shaped like a paraboloid of revolution. A light source is located 1 foot from the base along the axis of symmetry. If the opening of the searchlight is 3 feet across, find the depth.\r\n\r\n67. An arch is in the shape of a parabola. It has a span of 100 feet and a maximum height of 20 feet. Find the equation of the parabola, and determine the height of the arch 40 feet from the center.\r\n\r\n69. An object is projected so as to follow a parabolic path given by [latex]y=-{x}^{2}+96x[\/latex], where [latex]x[\/latex] is the horizontal distance traveled in feet and [latex]y[\/latex] is the height. Determine the maximum height the object reaches.\r\n

Rotation of Axes<\/h1>\r\n1. What effect does the [latex]xy[\/latex] term have on the graph of a conic section?\r\n\r\n3. If the equation of a conic section is written in the form [latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex], and [latex]{B}^{2}-4AC>0[\/latex], what can we conclude?\r\n\r\n5. For the equation [latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex], the value of [latex]\\theta [\/latex] that satisfies [latex]\\cot \\left(2\\theta \\right)=\\frac{A-C}{B}[\/latex] gives us what information?\r\n\r\nFor the following exercises, determine which conic section is represented based on the given equation.\r\n\r\n7. [latex]{x}^{2}-10x+4y - 10=0[\/latex]\r\n\r\n9. [latex]4{x}^{2}-{y}^{2}+8x - 1=0[\/latex]\r\n\r\n11. [latex]2{x}^{2}+3{y}^{2}-8x - 12y+2=0[\/latex]\r\n\r\n13. [latex]3{x}^{2}+6xy+3{y}^{2}-36y - 125=0[\/latex]\r\n\r\n15. [latex]2{x}^{2}+4\\sqrt{3}xy+6{y}^{2}-6x - 3=0[\/latex]\r\n\r\n17. [latex]8{x}^{2}+4\\sqrt{2}xy+4{y}^{2}-10x+1=0[\/latex]\r\n\r\nFor the following exercises, find a new representation of the given equation after rotating through the given angle.\r\n\r\n19. [latex]4{x}^{2}-xy+4{y}^{2}-2=0,\\theta =45^\\circ [\/latex]\r\n\r\n21. [latex]-2{x}^{2}+8xy+1=0,\\theta =45^\\circ [\/latex]\r\n\r\nFor the following exercises, determine the angle [latex]\\theta [\/latex] that will eliminate the [latex]xy[\/latex] term and write the corresponding equation without the [latex]xy[\/latex] term.\r\n\r\n23. [latex]{x}^{2}+3\\sqrt{3}xy+4{y}^{2}+y - 2=0[\/latex]\r\n\r\n25. [latex]9{x}^{2}-3\\sqrt{3}xy+6{y}^{2}+4y - 3=0[\/latex]\r\n\r\n27. [latex]16{x}^{2}+24xy+9{y}^{2}+6x - 6y+2=0[\/latex]\r\n\r\n29. [latex]{x}^{2}+4xy+{y}^{2}-2x+1=0[\/latex]\r\n\r\nFor the following exercises, rotate through the given angle based on the given equation. Give the new equation and graph the original and rotated equation.\r\n\r\n31. [latex]y=-{x}^{2},\\theta =-{45}^{\\circ }[\/latex]\r\n\r\n33. [latex]\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{1}=1,\\theta ={45}^{\\circ }[\/latex]\r\n\r\n35. [latex]{y}^{2}-{x}^{2}=1,\\theta ={45}^{\\circ }[\/latex]\r\n\r\n37. [latex]x={\\left(y - 1\\right)}^{2},\\theta ={30}^{\\circ }[\/latex]\r\n\r\nFor the following exercises, graph the equation relative to the [latex]\\begin{align}{x}^{\\prime }{y}^{\\prime }\\end{align}[\/latex] system in which the equation has no [latex]\\begin{align}{x}^{\\prime }{y}^{\\prime }\\end{align}[\/latex] term.\r\n\r\n39. [latex]xy=9[\/latex]\r\n\r\n41. [latex]{x}^{2}-10xy+{y}^{2}-24=0[\/latex]\r\n\r\n43. [latex]6{x}^{2}+2\\sqrt{3}xy+4{y}^{2}-21=0[\/latex]\r\n\r\n45. [latex]21{x}^{2}+2\\sqrt{3}xy+19{y}^{2}-18=0[\/latex]\r\n\r\n47. [latex]16{x}^{2}+24xy+9{y}^{2}-60x+80y=0[\/latex]\r\n\r\n49. [latex]4{x}^{2}-4xy+{y}^{2}-8\\sqrt{5}x - 16\\sqrt{5}y=0[\/latex]\r\n\r\nFor the following exercises, determine the angle of rotation in order to eliminate the [latex]xy[\/latex] term. Then graph the new set of axes.\r\n\r\n51. [latex]6{x}^{2}-5xy+6{y}^{2}+20x-y=0[\/latex]\r\n\r\n53. [latex]4{x}^{2}+6\\sqrt{3}xy+10{y}^{2}+20x - 40y=0[\/latex]\r\n\r\n55. [latex]16{x}^{2}+24xy+9{y}^{2}+20x - 44y=0[\/latex]\r\n\r\nFor the following exercises, determine the value of [latex]k[\/latex] based on the given equation.\r\n\r\n57. Given [latex]2{x}^{2}+kxy+12{y}^{2}+10x - 16y+28=0[\/latex], find [latex]k[\/latex] for the graph to be an ellipse.\r\n\r\n59. Given [latex]k{x}^{2}+8xy+8{y}^{2}-12x+16y+18=0[\/latex], find [latex]k[\/latex] for the graph to be a parabola.\r\n

Conic Sections in Polar Coordinates<\/h1>\r\n1. Explain how eccentricity determines which conic section is given.\r\n\r\nFor the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity.\r\n\r\n7. [latex]r=\\frac{3}{4 - 4\\text{ }\\sin \\text{ }\\theta }[\/latex]\r\n\r\n9. [latex]r=\\frac{5}{1+2\\text{ }\\sin \\text{ }\\theta }[\/latex]\r\n\r\n11. [latex]r=\\frac{3}{10+10\\text{ }\\cos \\text{ }\\theta }[\/latex]\r\n\r\n13. [latex]r=\\frac{4}{7+2\\text{ }\\cos \\text{ }\\theta }[\/latex]\r\n\r\n15. [latex]r\\left(3+5\\sin \\text{ }\\theta \\right)=11[\/latex]\r\n\r\n17. [latex]r\\left(7+8\\cos \\text{ }\\theta \\right)=7[\/latex]\r\n\r\nFor the following exercises, convert the polar equation of a conic section to a rectangular equation.\r\n\r\n19. [latex]r=\\frac{2}{5 - 3\\text{ }\\sin \\text{ }\\theta }[\/latex]\r\n\r\n21. [latex]r=\\frac{3}{2+5\\text{ }\\cos \\text{ }\\theta }[\/latex]\r\n\r\n23. [latex]r=\\frac{3}{8 - 8\\text{ }\\cos \\text{ }\\theta }[\/latex]\r\n\r\n25. [latex]r=\\frac{5}{5 - 11\\text{ }\\sin \\text{ }\\theta }[\/latex]\r\n\r\n27. [latex]r\\left(2-\\cos \\text{ }\\theta \\right)=1[\/latex]\r\n\r\n29. [latex]r=\\frac{6\\sec \\text{ }\\theta }{-2+3\\text{ }\\sec \\text{ }\\theta }[\/latex]\r\n\r\nFor the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci.\r\n\r\n31. [latex]r=\\frac{5}{2+\\cos \\text{ }\\theta }[\/latex]\r\n\r\n33. [latex]r=\\frac{10}{5 - 4\\text{ }\\sin \\text{ }\\theta }[\/latex]\r\n\r\n35. [latex]r=\\frac{8}{4 - 5\\text{ }\\cos \\text{ }\\theta }[\/latex]\r\n\r\n37. [latex]r=\\frac{2}{1-\\sin \\text{ }\\theta }[\/latex]\r\n\r\n39. [latex]r\\left(1+\\cos \\text{ }\\theta \\right)=5[\/latex]\r\n\r\n41. [latex]r\\left(3 - 2\\sin \\text{ }\\theta \\right)=6[\/latex]\r\n\r\nFor the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.\r\n\r\n43. Directrix: [latex]x=4;e=\\frac{1}{5}[\/latex]\r\n\r\n45. Directrix: [latex]y=2;e=2[\/latex]\r\n\r\n47. Directrix: [latex]x=1;e=1[\/latex]\r\n\r\n49. Directrix: [latex]x=-\\frac{1}{4};e=\\frac{7}{2}[\/latex]\r\n\r\n51. Directrix: [latex]y=4;e=\\frac{3}{2}[\/latex]\r\n\r\n53. Directrix: [latex]x=-5;e=\\frac{3}{4}[\/latex]\r\n\r\n55. Directrix: [latex]x=-3;e=\\frac{1}{3}[\/latex]\r\n\r\nEquations of conics with an [latex]xy[\/latex] term have rotated graphs. For the following exercises, express each equation in polar form with [latex]r[\/latex] as a function of [latex]\\theta [\/latex].\r\n\r\n57. [latex]{x}^{2}+xy+{y}^{2}=4[\/latex]\r\n\r\n59. [latex]16{x}^{2}+24xy+9{y}^{2}=4[\/latex]","rendered":"

The Ellipse<\/h1>\n

1. Define an ellipse in terms of its foci.<\/p>\n

3. What special case of the ellipse do we have when the major and minor axis are of the same length?<\/p>\n

5. What can be said about the symmetry of the graph of an ellipse with center at the origin and foci along the y-axis?<\/p>\n

For the following exercises, write the equation of an ellipse in standard form, and identify the end points of the major and minor axes as well as the foci.<\/p>\n

11. [latex]\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{49}=1[\/latex]<\/p>\n

13. [latex]{x}^{2}+9{y}^{2}=1[\/latex]<\/p>\n

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

17. [latex]\\frac{{\\left(x+5\\right)}^{2}}{4}+\\frac{{\\left(y - 7\\right)}^{2}}{9}=1[\/latex]<\/p>\n

19. [latex]4{x}^{2}-8x+9{y}^{2}-72y+112=0[\/latex]<\/p>\n

23. [latex]4{x}^{2}+40x+25{y}^{2}-100y+100=0[\/latex]<\/p>\n

For the following exercises, find the foci for the given ellipses.<\/p>\n

27. [latex]\\frac{{\\left(x+3\\right)}^{2}}{25}+\\frac{{\\left(y+1\\right)}^{2}}{36}=1[\/latex]<\/p>\n

29. [latex]{x}^{2}+{y}^{2}=1[\/latex]<\/p>\n

31. [latex]10{x}^{2}+{y}^{2}+200x=0[\/latex]<\/p>\n

For the following exercises, graph the given ellipses, noting center, vertices, and foci.<\/p>\n

33. [latex]\\frac{{x}^{2}}{16}+\\frac{{y}^{2}}{9}=1[\/latex]<\/p>\n

35. [latex]81{x}^{2}+49{y}^{2}=1[\/latex]<\/p>\n

37. [latex]\\frac{{\\left(x+3\\right)}^{2}}{9}+\\frac{{\\left(y - 3\\right)}^{2}}{9}=1[\/latex]<\/p>\n

39. [latex]4{x}^{2}-8x+16{y}^{2}-32y - 44=0[\/latex]<\/p>\n

41. [latex]{x}^{2}+8x+4{y}^{2}-40y+112=0[\/latex]<\/p>\n

45. [latex]4{x}^{2}+16x+4{y}^{2}+16y+16=0[\/latex]<\/p>\n

For the following exercises, use the given information about the graph of each ellipse to determine its equation.<\/p>\n

47. Center at the origin, symmetric with respect to the x<\/em>– and y<\/em>-axes, focus at [latex]\\left(0,-2\\right)[\/latex], and point on graph [latex]\\left(5,0\\right)[\/latex].<\/p>\n

49. Center [latex]\\left(4,2\\right)[\/latex] ; vertex [latex]\\left(9,2\\right)[\/latex] ; one focus: [latex]\\left(4+2\\sqrt{6},2\\right)[\/latex] .<\/p>\n

For the following exercises, given the graph of the ellipse, determine its equation.<\/p>\n

53.
\n\"\"<\/p>\n

55.
\n\"\"<\/p>\n

63. Find the equation of the ellipse that will just fit inside a box that is four times as wide as it is high. Express in terms of [latex]h[\/latex], the height.<\/p>\n

65. An arch has the shape of a semi-ellipse. The arch has a height of 12 feet and a span of 40 feet. Find an equation for the ellipse, and use that to find the distance from the center to a point at which the height is 6 feet. Round to the nearest hundredth.<\/p>\n

67. A person in a whispering gallery standing at one focus of the ellipse can whisper and be heard by a person standing at the other focus because all the sound waves that reach the ceiling are reflected to the other person. If a whispering gallery has a length of 120 feet, and the foci are located 30 feet from the center, find the height of the ceiling at the center.<\/p>\n

The Hyperbola<\/h1>\n

5. Where must the center of hyperbola be relative to its foci?<\/p>\n

For the following exercises, write the equation for the hyperbola in standard form if it is not already, and identify the vertices and foci, and write equations of asymptotes.<\/p>\n

11. [latex]\\frac{{x}^{2}}{25}-\\frac{{y}^{2}}{36}=1[\/latex]<\/p>\n

13. [latex]\\frac{{y}^{2}}{4}-\\frac{{x}^{2}}{81}=1[\/latex]<\/p>\n

15. [latex]\\frac{{\\left(x - 1\\right)}^{2}}{9}-\\frac{{\\left(y - 2\\right)}^{2}}{16}=1[\/latex]<\/p>\n

17. [latex]\\frac{{\\left(x - 2\\right)}^{2}}{49}-\\frac{{\\left(y+7\\right)}^{2}}{49}=1[\/latex]<\/p>\n

19. [latex]-9{x}^{2}-54x+9{y}^{2}-54y+81=0[\/latex]<\/p>\n

23. [latex]{x}^{2}+2x - 100{y}^{2}-1000y+2401=0[\/latex]<\/p>\n

For the following exercises, find the equations of the asymptotes for each hyperbola.<\/p>\n

27.\u00a0[latex]\\frac{{\\left(x - 3\\right)}^{2}}{{5}^{2}}-\\frac{{\\left(y+4\\right)}^{2}}{{2}^{2}}=1[\/latex]<\/p>\n

29.\u00a0[latex]9{x}^{2}-18x - 16{y}^{2}+32y - 151=0[\/latex]<\/p>\n

For the following exercises, sketch a graph of the hyperbola, labeling vertices and foci.<\/p>\n

31. [latex]\\frac{{x}^{2}}{49}-\\frac{{y}^{2}}{16}=1[\/latex]<\/p>\n

33. [latex]\\frac{{y}^{2}}{9}-\\frac{{x}^{2}}{25}=1[\/latex]<\/p>\n

35. [latex]\\frac{{\\left(y+5\\right)}^{2}}{9}-\\frac{{\\left(x - 4\\right)}^{2}}{25}=1[\/latex]<\/p>\n

37. [latex]\\frac{{\\left(y - 3\\right)}^{2}}{9}-\\frac{{\\left(x - 3\\right)}^{2}}{9}=1[\/latex]<\/p>\n

39. [latex]{x}^{2}-8x - 25{y}^{2}-100y - 109=0[\/latex]<\/p>\n

41. [latex]64{x}^{2}+128x - 9{y}^{2}-72y - 656=0[\/latex]<\/p>\n

43. [latex]-100{x}^{2}+1000x+{y}^{2}-10y - 2575=0[\/latex]<\/p>\n

For the following exercises, given information about the graph of the hyperbola, find its equation.<\/p>\n

45. Vertices at [latex]\\left(3,0\\right)[\/latex] and [latex]\\left(-3,0\\right)[\/latex] and one focus at [latex]\\left(5,0\\right)[\/latex].<\/p>\n

47. Vertices at [latex]\\left(1,1\\right)[\/latex] and [latex]\\left(11,1\\right)[\/latex] and one focus at [latex]\\left(12,1\\right)[\/latex].<\/p>\n

49. Center: [latex]\\left(4,2\\right)[\/latex]; vertex: [latex]\\left(9,2\\right)[\/latex]; one focus: [latex]\\left(4+\\sqrt{26},2\\right)[\/latex].<\/p>\n

For the following exercises, given the graph of the hyperbola, find its equation.<\/p>\n

51.
\n\"\"<\/p>\n

53.
\n\"\"<\/p>\n

55.
\n\"\"<\/p>\n

For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph.<\/p>\n

61. The hedge will follow the asymptotes [latex]y=x\\text{ and }y=-x[\/latex], and its closest distance to the center fountain is 5 yards.<\/p>\n

63. The hedge will follow the asymptotes [latex]y=\\frac{1}{2}x[\/latex] and [latex]y=-\\frac{1}{2}x[\/latex], and its closest distance to the center fountain is 10 yards.<\/p>\n

65. The hedge will follow the asymptotes [latex]\\text{ }y=\\frac{3}{4}x\\text{ and }y=-\\frac{3}{4}x[\/latex], and its closest distance to the center fountain is 20 yards.<\/p>\n

For the following exercises, assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the object’s path. Give the equation of the flight path of each object using the given information.<\/p>\n

67. The object enters along a path approximated by the line [latex]y=2x - 2[\/latex] and passes within 0.5 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line [latex]y=-2x+2[\/latex].<\/p>\n

69. The object enters along a path approximated by the line [latex]y=\\frac{1}{3}x - 1[\/latex] and passes within 1 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line [latex]\\text{ }y=-\\frac{1}{3}x+1[\/latex].<\/p>\n

The Parabola<\/h1>\n

1. Define a parabola in terms of its focus and directrix.<\/p>\n

3. If the equation of a parabola is written in standard form and [latex]p[\/latex] is negative and the directrix is a horizontal line, then what can we conclude about its graph?<\/p>\n

5. As the graph of a parabola becomes wider, what will happen to the distance between the focus and directrix?<\/p>\n

For the following exercises, rewrite the given equation in standard form, and then determine the vertex [latex]\\left(V\\right)[\/latex], focus [latex]\\left(F\\right)[\/latex], and directrix [latex]\\text{ }\\left(d\\right)\\text{ }[\/latex] of the parabola.<\/p>\n

11. [latex]x=8{y}^{2}[\/latex]<\/p>\n

13. [latex]y=-4{x}^{2}[\/latex]<\/p>\n

19. [latex]{\\left(y - 4\\right)}^{2}=2\\left(x+3\\right)[\/latex]<\/p>\n

21. [latex]{\\left(x+4\\right)}^{2}=24\\left(y+1\\right)[\/latex]<\/p>\n

23. [latex]{y}^{2}+12x - 6y+21=0[\/latex]<\/p>\n

25. [latex]5{x}^{2}-50x - 4y+113=0[\/latex]<\/p>\n

27. [latex]{x}^{2}-4x+2y - 6=0[\/latex]<\/p>\n

29. [latex]3{y}^{2}-4x - 6y+23=0[\/latex]<\/p>\n

For the following exercises, graph the parabola, labeling the focus and the directrix.<\/p>\n

31. [latex]x=\\frac{1}{8}{y}^{2}[\/latex]<\/p>\n

33. [latex]y=\\frac{1}{36}{x}^{2}[\/latex]<\/p>\n

35. [latex]{\\left(y - 2\\right)}^{2}=-\\frac{4}{3}\\left(x+2\\right)[\/latex]<\/p>\n

37. [latex]-6{\\left(y+5\\right)}^{2}=4\\left(x - 4\\right)[\/latex]<\/p>\n

39. [latex]{x}^{2}+8x+4y+20=0[\/latex]<\/p>\n

41. [latex]{y}^{2}-8x+10y+9=0[\/latex]<\/p>\n

43. [latex]{y}^{2}+2y - 12x+61=0[\/latex]<\/p>\n

For the following exercises, find the equation of the parabola given information about its graph.<\/p>\n

45. Vertex is [latex]\\left(0,0\\right)[\/latex]; directrix is [latex]y=4[\/latex], focus is [latex]\\left(0,-4\\right)[\/latex].<\/p>\n

47. Vertex is [latex]\\left(2,2\\right)[\/latex]; directrix is [latex]x=2-\\sqrt{2}[\/latex], focus is [latex]\\left(2+\\sqrt{2},2\\right)[\/latex].<\/p>\n

49. Vertex is [latex]\\left(\\sqrt{2},-\\sqrt{3}\\right)[\/latex]; directrix is [latex]x=2\\sqrt{2}[\/latex], focus is [latex]\\left(0,-\\sqrt{3}\\right)[\/latex].<\/p>\n

For the following exercises, determine the equation for the parabola from its graph.<\/p>\n

51.
\n\"\"<\/p>\n

53.
\n\"\"<\/p>\n

55.
\n\"\"<\/p>\n

61.\u00a0The mirror in an automobile headlight has a parabolic cross-section with the light bulb at the focus. On a schematic, the equation of the parabola is given as [latex]{x}^{2}=4y[\/latex]. At what coordinates should you place the light bulb?<\/p>\n

63. A satellite dish is shaped like a paraboloid of revolution. This means that it can be formed by rotating a parabola around its axis of symmetry. The receiver is to be located at the focus. If the dish is 12 feet across at its opening and 4 feet deep at its center, where should the receiver be placed?<\/p>\n

65. A searchlight is shaped like a paraboloid of revolution. A light source is located 1 foot from the base along the axis of symmetry. If the opening of the searchlight is 3 feet across, find the depth.<\/p>\n

67. An arch is in the shape of a parabola. It has a span of 100 feet and a maximum height of 20 feet. Find the equation of the parabola, and determine the height of the arch 40 feet from the center.<\/p>\n

69. An object is projected so as to follow a parabolic path given by [latex]y=-{x}^{2}+96x[\/latex], where [latex]x[\/latex] is the horizontal distance traveled in feet and [latex]y[\/latex] is the height. Determine the maximum height the object reaches.<\/p>\n

Rotation of Axes<\/h1>\n

1. What effect does the [latex]xy[\/latex] term have on the graph of a conic section?<\/p>\n

3. If the equation of a conic section is written in the form [latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex], and [latex]{B}^{2}-4AC>0[\/latex], what can we conclude?<\/p>\n

5. For the equation [latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex], the value of [latex]\\theta[\/latex] that satisfies [latex]\\cot \\left(2\\theta \\right)=\\frac{A-C}{B}[\/latex] gives us what information?<\/p>\n

For the following exercises, determine which conic section is represented based on the given equation.<\/p>\n

7. [latex]{x}^{2}-10x+4y - 10=0[\/latex]<\/p>\n

9. [latex]4{x}^{2}-{y}^{2}+8x - 1=0[\/latex]<\/p>\n

11. [latex]2{x}^{2}+3{y}^{2}-8x - 12y+2=0[\/latex]<\/p>\n

13. [latex]3{x}^{2}+6xy+3{y}^{2}-36y - 125=0[\/latex]<\/p>\n

15. [latex]2{x}^{2}+4\\sqrt{3}xy+6{y}^{2}-6x - 3=0[\/latex]<\/p>\n

17. [latex]8{x}^{2}+4\\sqrt{2}xy+4{y}^{2}-10x+1=0[\/latex]<\/p>\n

For the following exercises, find a new representation of the given equation after rotating through the given angle.<\/p>\n

19. [latex]4{x}^{2}-xy+4{y}^{2}-2=0,\\theta =45^\\circ[\/latex]<\/p>\n

21. [latex]-2{x}^{2}+8xy+1=0,\\theta =45^\\circ[\/latex]<\/p>\n

For the following exercises, determine the angle [latex]\\theta[\/latex] that will eliminate the [latex]xy[\/latex] term and write the corresponding equation without the [latex]xy[\/latex] term.<\/p>\n

23. [latex]{x}^{2}+3\\sqrt{3}xy+4{y}^{2}+y - 2=0[\/latex]<\/p>\n

25. [latex]9{x}^{2}-3\\sqrt{3}xy+6{y}^{2}+4y - 3=0[\/latex]<\/p>\n

27. [latex]16{x}^{2}+24xy+9{y}^{2}+6x - 6y+2=0[\/latex]<\/p>\n

29. [latex]{x}^{2}+4xy+{y}^{2}-2x+1=0[\/latex]<\/p>\n

For the following exercises, rotate through the given angle based on the given equation. Give the new equation and graph the original and rotated equation.<\/p>\n

31. [latex]y=-{x}^{2},\\theta =-{45}^{\\circ }[\/latex]<\/p>\n

33. [latex]\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{1}=1,\\theta ={45}^{\\circ }[\/latex]<\/p>\n

35. [latex]{y}^{2}-{x}^{2}=1,\\theta ={45}^{\\circ }[\/latex]<\/p>\n

37. [latex]x={\\left(y - 1\\right)}^{2},\\theta ={30}^{\\circ }[\/latex]<\/p>\n

For the following exercises, graph the equation relative to the [latex]\\begin{align}{x}^{\\prime }{y}^{\\prime }\\end{align}[\/latex] system in which the equation has no [latex]\\begin{align}{x}^{\\prime }{y}^{\\prime }\\end{align}[\/latex] term.<\/p>\n

39. [latex]xy=9[\/latex]<\/p>\n

41. [latex]{x}^{2}-10xy+{y}^{2}-24=0[\/latex]<\/p>\n

43. [latex]6{x}^{2}+2\\sqrt{3}xy+4{y}^{2}-21=0[\/latex]<\/p>\n

45. [latex]21{x}^{2}+2\\sqrt{3}xy+19{y}^{2}-18=0[\/latex]<\/p>\n

47. [latex]16{x}^{2}+24xy+9{y}^{2}-60x+80y=0[\/latex]<\/p>\n

49. [latex]4{x}^{2}-4xy+{y}^{2}-8\\sqrt{5}x - 16\\sqrt{5}y=0[\/latex]<\/p>\n

For the following exercises, determine the angle of rotation in order to eliminate the [latex]xy[\/latex] term. Then graph the new set of axes.<\/p>\n

51. [latex]6{x}^{2}-5xy+6{y}^{2}+20x-y=0[\/latex]<\/p>\n

53. [latex]4{x}^{2}+6\\sqrt{3}xy+10{y}^{2}+20x - 40y=0[\/latex]<\/p>\n

55. [latex]16{x}^{2}+24xy+9{y}^{2}+20x - 44y=0[\/latex]<\/p>\n

For the following exercises, determine the value of [latex]k[\/latex] based on the given equation.<\/p>\n

57. Given [latex]2{x}^{2}+kxy+12{y}^{2}+10x - 16y+28=0[\/latex], find [latex]k[\/latex] for the graph to be an ellipse.<\/p>\n

59. Given [latex]k{x}^{2}+8xy+8{y}^{2}-12x+16y+18=0[\/latex], find [latex]k[\/latex] for the graph to be a parabola.<\/p>\n

Conic Sections in Polar Coordinates<\/h1>\n

1. Explain how eccentricity determines which conic section is given.<\/p>\n

For the following exercises, identify the conic with a focus at the origin, and then give the directrix and eccentricity.<\/p>\n

7. [latex]r=\\frac{3}{4 - 4\\text{ }\\sin \\text{ }\\theta }[\/latex]<\/p>\n

9. [latex]r=\\frac{5}{1+2\\text{ }\\sin \\text{ }\\theta }[\/latex]<\/p>\n

11. [latex]r=\\frac{3}{10+10\\text{ }\\cos \\text{ }\\theta }[\/latex]<\/p>\n

13. [latex]r=\\frac{4}{7+2\\text{ }\\cos \\text{ }\\theta }[\/latex]<\/p>\n

15. [latex]r\\left(3+5\\sin \\text{ }\\theta \\right)=11[\/latex]<\/p>\n

17. [latex]r\\left(7+8\\cos \\text{ }\\theta \\right)=7[\/latex]<\/p>\n

For the following exercises, convert the polar equation of a conic section to a rectangular equation.<\/p>\n

19. [latex]r=\\frac{2}{5 - 3\\text{ }\\sin \\text{ }\\theta }[\/latex]<\/p>\n

21. [latex]r=\\frac{3}{2+5\\text{ }\\cos \\text{ }\\theta }[\/latex]<\/p>\n

23. [latex]r=\\frac{3}{8 - 8\\text{ }\\cos \\text{ }\\theta }[\/latex]<\/p>\n

25. [latex]r=\\frac{5}{5 - 11\\text{ }\\sin \\text{ }\\theta }[\/latex]<\/p>\n

27. [latex]r\\left(2-\\cos \\text{ }\\theta \\right)=1[\/latex]<\/p>\n

29. [latex]r=\\frac{6\\sec \\text{ }\\theta }{-2+3\\text{ }\\sec \\text{ }\\theta }[\/latex]<\/p>\n

For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci.<\/p>\n

31. [latex]r=\\frac{5}{2+\\cos \\text{ }\\theta }[\/latex]<\/p>\n

33. [latex]r=\\frac{10}{5 - 4\\text{ }\\sin \\text{ }\\theta }[\/latex]<\/p>\n

35. [latex]r=\\frac{8}{4 - 5\\text{ }\\cos \\text{ }\\theta }[\/latex]<\/p>\n

37. [latex]r=\\frac{2}{1-\\sin \\text{ }\\theta }[\/latex]<\/p>\n

39. [latex]r\\left(1+\\cos \\text{ }\\theta \\right)=5[\/latex]<\/p>\n

41. [latex]r\\left(3 - 2\\sin \\text{ }\\theta \\right)=6[\/latex]<\/p>\n

For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix.<\/p>\n

43. Directrix: [latex]x=4;e=\\frac{1}{5}[\/latex]<\/p>\n

45. Directrix: [latex]y=2;e=2[\/latex]<\/p>\n

47. Directrix: [latex]x=1;e=1[\/latex]<\/p>\n

49. Directrix: [latex]x=-\\frac{1}{4};e=\\frac{7}{2}[\/latex]<\/p>\n

51. Directrix: [latex]y=4;e=\\frac{3}{2}[\/latex]<\/p>\n

53. Directrix: [latex]x=-5;e=\\frac{3}{4}[\/latex]<\/p>\n

55. Directrix: [latex]x=-3;e=\\frac{1}{3}[\/latex]<\/p>\n

Equations of conics with an [latex]xy[\/latex] term have rotated graphs. For the following exercises, express each equation in polar form with [latex]r[\/latex] as a function of [latex]\\theta[\/latex].<\/p>\n

57. [latex]{x}^{2}+xy+{y}^{2}=4[\/latex]<\/p>\n

59. [latex]16{x}^{2}+24xy+9{y}^{2}=4[\/latex]<\/p>\n","protected":false},"author":67,"menu_order":33,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":522,"module-header":"practice","content_attributions":[],"internal_book_links":[],"video_content":null,"cc_video_embed_content":{"cc_scripts":"","media_targets":[]},"try_it_collection":null,"_links":{"self":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2373"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/users\/67"}],"version-history":[{"count":6,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2373\/revisions"}],"predecessor-version":[{"id":5722,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2373\/revisions\/5722"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/parts\/522"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapters\/2373\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/media?parent=2373"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/pressbooks\/v2\/chapter-type?post=2373"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/contributor?post=2373"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/precalculus\/wp-json\/wp\/v2\/license?post=2373"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}