{"id":2575,"date":"2024-08-07T20:29:40","date_gmt":"2024-08-07T20:29:40","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=2575"},"modified":"2025-08-15T16:41:16","modified_gmt":"2025-08-15T16:41:16","slug":"ellipses-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/ellipses-learn-it-2\/","title":{"raw":"Ellipses: Learn It 2","rendered":"Ellipses: Learn It 2"},"content":{"raw":"

Writing Equations and Graphing Ellipses Centered at the Origin in Standard Form<\/h2>\r\n
\r\n
\r\n
\r\n\r\nThe key features of the\u00a0ellipse<\/span>\u00a0are its center,\u00a0vertices<\/span>,\u00a0co-vertices<\/span>,\u00a0foci<\/span>, and lengths and positions of the\u00a0major and minor axes<\/span>. Just as with other equations, we can identify all of these features just by looking at the standard form of the equation. There are four variations of the standard form of the ellipse. These variations are categorized first by the location of the center (the origin or not the origin), and then by the position (horizontal or vertical). Each is presented along with a description of how the parts of the equation relate to the graph. Interpreting these parts allows us to form a mental picture of the ellipse.\r\n\r\n
\r\n

standard forms of the equation of an ellipse with center [latex](0,0)[\/latex]<\/h3>\r\nThe standard form of the equation of an ellipse with center [latex]\\left(0,0\\right)[\/latex] and major axis<\/strong> parallel to the [latex]x[\/latex]-axis is\r\n

[latex]\\dfrac{{x}^{2}}{{a}^{2}}+\\dfrac{{y}^{2}}{{b}^{2}}=1[\/latex]<\/p>\r\n \r\n\r\nwhere:\r\n\r\n\"\"\r\n

    \r\n \t
  • [latex]a>b[\/latex]<\/li>\r\n \t
  • the length of the major axis is [latex]2a[\/latex]<\/li>\r\n \t
  • the coordinates of the vertices are [latex]\\left(\\pm a,0\\right)[\/latex]<\/li>\r\n \t
  • the length of the minor axis is [latex]2b[\/latex]<\/li>\r\n \t
  • the coordinates of the co-vertices are [latex]\\left(0,\\pm b\\right)[\/latex]<\/li>\r\n \t
  • the coordinates of the foci are [latex]\\left(\\pm c,0\\right)[\/latex], where [latex]{c}^{2}={a}^{2}-{b}^{2}[\/latex].<\/li>\r\n<\/ul>\r\nThe standard form of the equation of an ellipse with center [latex]\\left(0,0\\right)[\/latex] and major axis parallel to the y-axis is\r\n\r\n
    [latex]\\dfrac{{x}^{2}}{{b}^{2}}+\\dfrac{{y}^{2}}{{a}^{2}}=1[\/latex]<\/center>\r\nwhere\r\n
      \r\n \t
    • \r\n\r\n[caption id=\"attachment_2566\" align=\"alignright\" width=\"200\"]\"\" Ellipse with the y-axis as the Major Axis[\/caption]\r\n\r\n[latex]a>b[\/latex]<\/li>\r\n \t
    • the length of the major axis is [latex]2a[\/latex]<\/li>\r\n \t
    • the coordinates of the vertices are [latex]\\left(0,\\pm a\\right)[\/latex]<\/li>\r\n \t
    • the length of the minor axis is [latex]2b[\/latex]<\/li>\r\n \t
    • the coordinates of the co-vertices are [latex]\\left(\\pm b,0\\right)[\/latex]<\/li>\r\n \t
    • the coordinates of the foci are [latex]\\left(0,\\pm c\\right)[\/latex], where [latex]{c}^{2}={a}^{2}-{b}^{2}[\/latex].<\/li>\r\n<\/ul>\r\n<\/section>
      How To: Given the vertices and foci of an ellipse centered at the origin, write its equation in standard form.<\/strong>\r\n
        \r\n \t
      1. Determine whether the major axis is on the [latex]x[\/latex]- or [latex]y[\/latex]-axis.\r\n
          \r\n \t
        1. If the given coordinates of the vertices and foci have the form [latex](\\pm a,0)[\/latex] and\u00a0[latex](\\pm c,0)[\/latex] respectively, then the major axis is parallel to the [latex]x[\/latex]-axis. Use the standard form [latex]\\dfrac{{x}^{2}}{{a}^{2}}+\\dfrac{{y}^{2}}{{b}^{2}}=1[\/latex].<\/li>\r\n \t
        2. If the given coordinates of the vertices and foci have the form [latex](0,\\pm a)[\/latex] and\u00a0[latex](0,\\pm c)[\/latex] respectively, then the major axis is parallel to the [latex]y[\/latex]-axis. Use the standard form [latex]\\dfrac{{x}^{2}}{{b}^{2}}+\\dfrac{{y}^{2}}{{a}^{2}}=1[\/latex].<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
        3. Use the equation [latex]c^2=a^2-b^2[\/latex] along with the given coordinates of the vertices and foci, to solve for [latex]b^2[\/latex].<\/li>\r\n \t
        4. Substitute the values for\u00a0[latex]a^2[\/latex] and\u00a0[latex]b^2[\/latex] into the standard form of the equation determined in Step 1.<\/li>\r\n<\/ol>\r\n<\/section>
          What is the standard form equation of the ellipse that has vertices [latex](\\pm 8,0)[\/latex] and foci\u00a0[latex](\\pm 5,0)[\/latex]?[reveal-answer q=\"796590\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"796590\"]The foci are on the\u00a0[latex]x[\/latex]-axis, so the major axis is the\u00a0[latex]x[\/latex]-axis. Thus the equation will have the form:\r\n

          [latex]\\dfrac{{x}^{2}}{{a}^{2}}+\\dfrac{{y}^{2}}{{b}^{2}}=1[\/latex]<\/p>\r\nThe vertices are\u00a0[latex](\\pm 8,0)[\/latex], so [latex]a=8[\/latex] and [latex]a^2=64[\/latex].\r\n\r\nThe foci are\u00a0[latex](\\pm 5,0)[\/latex], so [latex]c=5[\/latex] and [latex]c^2=25[\/latex].\r\n\r\nWe know that the vertices and foci are related by the equation\u00a0[latex]c^2=a^2-b^2[\/latex]. Solving for [latex]b^2[\/latex] we have\r\n

          [latex]\\begin{align}&c^2=a^2-b^2&& \\\\ &25 = 64 - b^2 && \\text{Substitute for }c^2 \\text{ and }a^2. \\\\ &b^2=39 && \\text{Solve for } b^2. \\end{align}[\/latex]<\/p>\r\nNow we need only substitute [latex]a^2 = 64[\/latex] and [latex]b^2=39[\/latex] into the standard form of the equation. The equation of the ellipse is\r\n

          [latex]\\dfrac{{x}^{2}}{64}+\\dfrac{{y}^{2}}{39}=1[\/latex]<\/p>\r\n[\/hidden-answer]\r\n\r\n<\/section>

          [ohm_question hide_question_numbers=1]291343[\/ohm_question]<\/section>Just as we can write the equation for an ellipse given its graph, we can graph an ellipse given its equation. To graph ellipses centered at the origin, we use the standard form [latex]\\dfrac{{x}^{2}}{{a}^{2}}+\\dfrac{{y}^{2}}{{b}^{2}}=1,\\text{ }a>b[\/latex] for horizontal ellipses and [latex]\\dfrac{{x}^{2}}{{b}^{2}}+\\dfrac{{y}^{2}}{{a}^{2}}=1,\\text{ }a>b[\/latex] for vertical ellipses.\r\n\r\n
          How To: Given the standard form of an equation for an ellipse centered at [latex]\\left(0,0\\right)[\/latex], sketch the graph.<\/strong>\r\n
            \r\n \t
          • Use the standard forms of the equations of an ellipse to determine the major axis, vertices, co-vertices, and foci.\r\n
              \r\n \t
            • If the equation is in the form [latex]\\dfrac{{x}^{2}}{{a}^{2}}+\\dfrac{{y}^{2}}{{b}^{2}}=1[\/latex], where [latex]a>b[\/latex], then\r\n
                \r\n \t
              • the major axis is the [latex]x[\/latex]-axis<\/li>\r\n \t
              • the coordinates of the vertices are [latex]\\left(\\pm a,0\\right)[\/latex]<\/li>\r\n \t
              • the coordinates of the co-vertices are [latex]\\left(0,\\pm b\\right)[\/latex]<\/li>\r\n \t
              • the coordinates of the foci are [latex]\\left(\\pm c,0\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
              • If the equation is in the form [latex]\\dfrac{{x}^{2}}{{b}^{2}}+\\dfrac{{y}^{2}}{{a}^{2}}=1[\/latex], where [latex]a>b[\/latex], then\r\n
                  \r\n \t
                • the major axis is the [latex]y[\/latex]-axis<\/li>\r\n \t
                • the coordinates of the vertices are [latex]\\left(0,\\pm a\\right)[\/latex]<\/li>\r\n \t
                • the coordinates of the co-vertices are [latex]\\left(\\pm b,0\\right)[\/latex]<\/li>\r\n \t
                • the coordinates of the foci are [latex]\\left(0,\\pm c\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
                • Solve for [latex]c[\/latex] using the equation [latex]{c}^{2}={a}^{2}-{b}^{2}[\/latex].<\/li>\r\n \t
                • Plot the center, vertices, co-vertices, and foci in the coordinate plane, and draw a smooth curve to form the ellipse.<\/li>\r\n<\/ul>\r\n<\/section>
                  Graph the ellipse given by the equation, [latex]\\dfrac{{x}^{2}}{9}+\\dfrac{{y}^{2}}{25}=1[\/latex]. Identify and label the center, vertices, co-vertices, and foci.[reveal-answer q=\"819216\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"819216\"]First, we determine the position of the major axis. Because [latex]25>9[\/latex], the major axis is on the [latex]y[\/latex]-axis. It is vertically oriented.Comparing the equation with the standard form:\r\n

                  [latex]a^2 = 25 \\text{ and }b^2 = 9[\/latex]<\/p>\r\nTherefore: [latex]a = \\sqrt{25} = 5[\/latex] and [latex]b = \\sqrt{9} = 3[\/latex].\r\n\r\nFirst, let's find the value of [latex]c[\/latex] to find the coordinates of the foci:\r\n

                  [latex]\\begin{align}c&=\\pm \\sqrt{{a}^{2}-{b}^{2}} \\\\ &=\\pm \\sqrt{25 - 9} \\\\ &=\\pm \\sqrt{16} \\\\ &=\\pm 4 \\end{align}[\/latex]<\/p>\r\nKey Components:<\/strong>\r\n\r\n[latex]\\begin{align*} &\\text{Center: } (0, 0) \\\\ &\\text{Vertices: } (0, \\pm 5) \\\\ &\\text{Co-vertices: } (\\pm 3, 0) \\\\ &\\text{Foci: } (0, \\pm 4) \\\\ \\end{align*}[\/latex]\r\n\r\nNext, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"351\"]\"\" Ellipse on a coordinate plane with the center and six points labeled[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

                  Graph the ellipse given by the equation [latex]4{x}^{2}+25{y}^{2}=100[\/latex]. Rewrite the equation in standard form. Then identify and label the center, vertices, co-vertices, and foci.[reveal-answer q=\"950605\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"950605\"]First, use algebra to rewrite the equation in standard form.\r\n

                  [latex]\\begin{gathered} 4{x}^{2}+25{y}^{2}=100 \\\\[1.5mm] \\dfrac{4{x}^{2}}{100}+\\dfrac{25{y}^{2}}{100}=\\dfrac{100}{100} \\\\[1.5mm] \\dfrac{{x}^{2}}{25}+\\dfrac{{y}^{2}}{4}=1 \\end{gathered}[\/latex]<\/p>\r\nNext, we determine the position of the major axis. Because [latex]25>4[\/latex], the major axis is on the [latex]x[\/latex]-axis. Therefore, the equation is in the form [latex]\\dfrac{{x}^{2}}{{a}^{2}}+\\dfrac{{y}^{2}}{{b}^{2}}=1[\/latex], where [latex]{a}^{2}=25[\/latex] and [latex]{b}^{2}=4[\/latex]. It follows that:\r\n

                    \r\n \t
                  • the center of the ellipse is [latex]\\left(0,0\\right)[\/latex]<\/li>\r\n \t
                  • the coordinates of the vertices are [latex]\\left(\\pm a,0\\right)=\\left(\\pm \\sqrt{25},0\\right)=\\left(\\pm 5,0\\right)[\/latex]<\/li>\r\n \t
                  • the coordinates of the co-vertices are [latex]\\left(0,\\pm b\\right)=\\left(0,\\pm \\sqrt{4}\\right)=\\left(0,\\pm 2\\right)[\/latex]<\/li>\r\n \t
                  • the coordinates of the foci are [latex]\\left(\\pm c,0\\right)[\/latex], where [latex]{c}^{2}={a}^{2}-{b}^{2}[\/latex]. Solving for [latex]c[\/latex], we have:<\/li>\r\n<\/ul>\r\n

                    [latex]\\begin{align}c&=\\pm \\sqrt{{a}^{2}-{b}^{2}} \\\\ &=\\pm \\sqrt{25 - 4} \\\\ &=\\pm \\sqrt{21} \\end{align}[\/latex]<\/p>\r\nTherefore the coordinates of the foci are [latex]\\left(\\pm \\sqrt{21},0\\right)[\/latex].\r\n\r\nNext, we plot and label the center, vertices, co-vertices, and foci, and draw a smooth curve to form the ellipse.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"731\"]\"\" Ellipse on a coordinate plane with the center and six points labeled[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>

                    [ohm2_question hide_question_numbers=1]24891[\/ohm2_question]<\/section>
                    [ohm2_question hide_question_numbers=1]24892[\/ohm2_question]<\/section><\/div>\r\n<\/div>\r\n<\/div>","rendered":"

                    Writing Equations and Graphing Ellipses Centered at the Origin in Standard Form<\/h2>\n
                    \n
                    \n
                    \n

                    The key features of the\u00a0ellipse<\/span>\u00a0are its center,\u00a0vertices<\/span>,\u00a0co-vertices<\/span>,\u00a0foci<\/span>, and lengths and positions of the\u00a0major and minor axes<\/span>. Just as with other equations, we can identify all of these features just by looking at the standard form of the equation. There are four variations of the standard form of the ellipse. These variations are categorized first by the location of the center (the origin or not the origin), and then by the position (horizontal or vertical). Each is presented along with a description of how the parts of the equation relate to the graph. Interpreting these parts allows us to form a mental picture of the ellipse.<\/p>\n

                    \n

                    standard forms of the equation of an ellipse with center [latex](0,0)[\/latex]<\/h3>\n

                    The standard form of the equation of an ellipse with center [latex]\\left(0,0\\right)[\/latex] and major axis<\/strong> parallel to the [latex]x[\/latex]-axis is<\/p>\n

                    [latex]\\dfrac{{x}^{2}}{{a}^{2}}+\\dfrac{{y}^{2}}{{b}^{2}}=1[\/latex]<\/p>\n

                     <\/p>\n

                    where:<\/p>\n

                    \"\"<\/p>\n

                      \n
                    • [latex]a>b[\/latex]<\/li>\n
                    • the length of the major axis is [latex]2a[\/latex]<\/li>\n
                    • the coordinates of the vertices are [latex]\\left(\\pm a,0\\right)[\/latex]<\/li>\n
                    • the length of the minor axis is [latex]2b[\/latex]<\/li>\n
                    • the coordinates of the co-vertices are [latex]\\left(0,\\pm b\\right)[\/latex]<\/li>\n
                    • the coordinates of the foci are [latex]\\left(\\pm c,0\\right)[\/latex], where [latex]{c}^{2}={a}^{2}-{b}^{2}[\/latex].<\/li>\n<\/ul>\n

                      The standard form of the equation of an ellipse with center [latex]\\left(0,0\\right)[\/latex] and major axis parallel to the y-axis is<\/p>\n

                      [latex]\\dfrac{{x}^{2}}{{b}^{2}}+\\dfrac{{y}^{2}}{{a}^{2}}=1[\/latex]<\/div>\n

                      where<\/p>\n

                        \n
                      • \n
                        \"\"
                        Ellipse with the y-axis as the Major Axis<\/figcaption><\/figure>\n

                        [latex]a>b[\/latex]<\/li>\n

                      • the length of the major axis is [latex]2a[\/latex]<\/li>\n
                      • the coordinates of the vertices are [latex]\\left(0,\\pm a\\right)[\/latex]<\/li>\n
                      • the length of the minor axis is [latex]2b[\/latex]<\/li>\n
                      • the coordinates of the co-vertices are [latex]\\left(\\pm b,0\\right)[\/latex]<\/li>\n
                      • the coordinates of the foci are [latex]\\left(0,\\pm c\\right)[\/latex], where [latex]{c}^{2}={a}^{2}-{b}^{2}[\/latex].<\/li>\n<\/ul>\n<\/section>\n
                        How To: Given the vertices and foci of an ellipse centered at the origin, write its equation in standard form.<\/strong><\/p>\n
                          \n
                        1. Determine whether the major axis is on the [latex]x[\/latex]– or [latex]y[\/latex]-axis.\n
                            \n
                          1. If the given coordinates of the vertices and foci have the form [latex](\\pm a,0)[\/latex] and\u00a0[latex](\\pm c,0)[\/latex] respectively, then the major axis is parallel to the [latex]x[\/latex]-axis. Use the standard form [latex]\\dfrac{{x}^{2}}{{a}^{2}}+\\dfrac{{y}^{2}}{{b}^{2}}=1[\/latex].<\/li>\n
                          2. If the given coordinates of the vertices and foci have the form [latex](0,\\pm a)[\/latex] and\u00a0[latex](0,\\pm c)[\/latex] respectively, then the major axis is parallel to the [latex]y[\/latex]-axis. Use the standard form [latex]\\dfrac{{x}^{2}}{{b}^{2}}+\\dfrac{{y}^{2}}{{a}^{2}}=1[\/latex].<\/li>\n<\/ol>\n<\/li>\n
                          3. Use the equation [latex]c^2=a^2-b^2[\/latex] along with the given coordinates of the vertices and foci, to solve for [latex]b^2[\/latex].<\/li>\n
                          4. Substitute the values for\u00a0[latex]a^2[\/latex] and\u00a0[latex]b^2[\/latex] into the standard form of the equation determined in Step 1.<\/li>\n<\/ol>\n<\/section>\n
                            What is the standard form equation of the ellipse that has vertices [latex](\\pm 8,0)[\/latex] and foci\u00a0[latex](\\pm 5,0)[\/latex]?<\/p>\n