{"id":1436,"date":"2023-06-05T14:51:43","date_gmt":"2023-06-05T14:51:43","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-the-ellipse\/"},"modified":"2023-06-05T14:51:43","modified_gmt":"2023-06-05T14:51:43","slug":"solutions-for-the-ellipse","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-the-ellipse\/","title":{"raw":"Solutions 65: The Ellipse","rendered":"Solutions 65: The Ellipse"},"content":{"raw":"\n<h2>Solutions to Odd-Numbered Exercises<\/h2>\n1.&nbsp;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.\n\n3.&nbsp;This special case would be a circle.\n\n5.&nbsp;It is symmetric about the <em>x<\/em>-axis, <em>y<\/em>-axis, and the origin.\n\n7.&nbsp;yes; [latex]\\frac{{x}^{2}}{{3}^{2}}+\\frac{{y}^{2}}{{2}^{2}}=1[\/latex]\n\n9.&nbsp;yes; [latex]\\frac{{x}^{2}}{{\\left(\\frac{1}{2}\\right)}^{2}}+\\frac{{y}^{2}}{{\\left(\\frac{1}{3}\\right)}^{2}}=1[\/latex]\n\n11.&nbsp;[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].\n\n13.&nbsp;[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].\n\n15.&nbsp;[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].\n\n17.&nbsp;[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].\n\n19.&nbsp;[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].\n\n21.&nbsp;[latex]\\frac{{\\left(x - 3\\right)}^{2}}{{\\left(3\\sqrt{2}\\right)}^{2}}+\\frac{{\\left(y - 5\\right)}^{2}}{{\\left(\\sqrt{2}\\right)}^{2}}=1[\/latex]; Endpoints of major axis [latex]\\left(3+3\\sqrt{2},5\\right),\\left(3 - 3\\sqrt{2},5\\right)[\/latex]. Endpoints of minor axis [latex]\\left(3,5+\\sqrt{2}\\right),\\left(3,5-\\sqrt{2}\\right)[\/latex]. Foci at [latex]\\left(7,5\\right),\\left(-1,5\\right)[\/latex].\n\n23.&nbsp;[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].\n\n25.&nbsp;[latex]\\frac{{\\left(x+3\\right)}^{2}}{{\\left(5\\right)}^{2}}+\\frac{{\\left(y+4\\right)}^{2}}{{\\left(2\\right)}^{2}}=1[\/latex]; Endpoints of major axis [latex]\\left(2,-4\\right),\\left(-8,-4\\right)[\/latex]. Endpoints of minor axis [latex]\\left(-3,-2\\right),\\left(-3,-6\\right)[\/latex]. Foci at [latex]\\left(-3+\\sqrt{21},-4\\right),\\left(-3-\\sqrt{21},-4\\right)[\/latex].\n\n27.&nbsp;Foci [latex]\\left(-3,-1+\\sqrt{11}\\right),\\left(-3,-1-\\sqrt{11}\\right)[\/latex]\n\n29.&nbsp;Focus [latex]\\left(0,0\\right)[\/latex]\n\n31.&nbsp;Foci [latex]\\left(-10,30\\right),\\left(-10,-30\\right)[\/latex]\n\n33.&nbsp;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]\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182748\/CNX_Precalc_Figure_10_01_202.jpg\" alt=\"\">\n\n35.&nbsp;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]\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182751\/CNX_Precalc_Figure_10_01_204.jpg\" alt=\"\">\n\n37.&nbsp;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]\nNote that this ellipse is a circle. The circle has only one focus, which coincides with the center.\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182752\/CNX_Precalc_Figure_10_01_206.jpg\" alt=\"\">\n\n39.&nbsp;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]\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182754\/CNX_Precalc_Figure_10_01_208.jpg\" alt=\"\">\n\n41.&nbsp;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]\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182757\/CNX_Precalc_Figure_10_01_210.jpg\" alt=\"\">\n\n43.&nbsp;Center [latex]\\left(-2,1\\right)[\/latex], Vertices [latex]\\left(0,1\\right),\\left(-4,1\\right),\\left(-2,5\\right),\\left(-2,-3\\right)[\/latex], Foci [latex]\\left(-2,1+2\\sqrt{3}\\right),\\left(-2,1 - 2\\sqrt{3}\\right)[\/latex]\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182759\/CNX_Precalc_Figure_10_01_212.jpg\" alt=\"\">\n\n45.&nbsp;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]\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182801\/CNX_Precalc_Figure_10_01_214.jpg\" alt=\"\">\n\n47.&nbsp;[latex]\\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{29}=1[\/latex]\n\n49.&nbsp;[latex]\\frac{{\\left(x - 4\\right)}^{2}}{25}+\\frac{{\\left(y - 2\\right)}^{2}}{1}=1[\/latex]\n\n51.&nbsp;[latex]\\frac{{\\left(x+3\\right)}^{2}}{16}+\\frac{{\\left(y - 4\\right)}^{2}}{4}=1[\/latex]\n\n53.&nbsp;[latex]\\frac{{x}^{2}}{81}+\\frac{{y}^{2}}{9}=1[\/latex]\n\n55.&nbsp;[latex]\\frac{{\\left(x+2\\right)}^{2}}{4}+\\frac{{\\left(y - 2\\right)}^{2}}{9}=1[\/latex]\n\n57.&nbsp;[latex]\\text{Area}=12\\pi[\/latex] square units\n\n59.&nbsp;[latex]\\text{Area}=2\\sqrt{5}\\pi[\/latex] square units\n\n61.&nbsp;[latex]\\text{Area }9\\pi[\/latex] square units\n\n63.&nbsp;[latex]\\frac{{x}^{2}}{4{h}^{2}}+\\frac{{y}^{2}}{\\frac{1}{4}{h}^{2}}=1[\/latex]\n\n65.&nbsp;[latex]\\frac{{x}^{2}}{400}+\\frac{{y}^{2}}{144}=1[\/latex]. Distance = 17.32 feet\n\n67.&nbsp;Approximately 51.96 feet\n","rendered":"<h2>Solutions to Odd-Numbered Exercises<\/h2>\n<p>1.&nbsp;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.<\/p>\n<p>3.&nbsp;This special case would be a circle.<\/p>\n<p>5.&nbsp;It is symmetric about the <em>x<\/em>-axis, <em>y<\/em>-axis, and the origin.<\/p>\n<p>7.&nbsp;yes; [latex]\\frac{{x}^{2}}{{3}^{2}}+\\frac{{y}^{2}}{{2}^{2}}=1[\/latex]<\/p>\n<p>9.&nbsp;yes; [latex]\\frac{{x}^{2}}{{\\left(\\frac{1}{2}\\right)}^{2}}+\\frac{{y}^{2}}{{\\left(\\frac{1}{3}\\right)}^{2}}=1[\/latex]<\/p>\n<p>11.&nbsp;[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].<\/p>\n<p>13.&nbsp;[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].<\/p>\n<p>15.&nbsp;[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].<\/p>\n<p>17.&nbsp;[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].<\/p>\n<p>19.&nbsp;[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].<\/p>\n<p>21.&nbsp;[latex]\\frac{{\\left(x - 3\\right)}^{2}}{{\\left(3\\sqrt{2}\\right)}^{2}}+\\frac{{\\left(y - 5\\right)}^{2}}{{\\left(\\sqrt{2}\\right)}^{2}}=1[\/latex]; Endpoints of major axis [latex]\\left(3+3\\sqrt{2},5\\right),\\left(3 - 3\\sqrt{2},5\\right)[\/latex]. Endpoints of minor axis [latex]\\left(3,5+\\sqrt{2}\\right),\\left(3,5-\\sqrt{2}\\right)[\/latex]. Foci at [latex]\\left(7,5\\right),\\left(-1,5\\right)[\/latex].<\/p>\n<p>23.&nbsp;[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].<\/p>\n<p>25.&nbsp;[latex]\\frac{{\\left(x+3\\right)}^{2}}{{\\left(5\\right)}^{2}}+\\frac{{\\left(y+4\\right)}^{2}}{{\\left(2\\right)}^{2}}=1[\/latex]; Endpoints of major axis [latex]\\left(2,-4\\right),\\left(-8,-4\\right)[\/latex]. Endpoints of minor axis [latex]\\left(-3,-2\\right),\\left(-3,-6\\right)[\/latex]. Foci at [latex]\\left(-3+\\sqrt{21},-4\\right),\\left(-3-\\sqrt{21},-4\\right)[\/latex].<\/p>\n<p>27.&nbsp;Foci [latex]\\left(-3,-1+\\sqrt{11}\\right),\\left(-3,-1-\\sqrt{11}\\right)[\/latex]<\/p>\n<p>29.&nbsp;Focus [latex]\\left(0,0\\right)[\/latex]<\/p>\n<p>31.&nbsp;Foci [latex]\\left(-10,30\\right),\\left(-10,-30\\right)[\/latex]<\/p>\n<p>33.&nbsp;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]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182748\/CNX_Precalc_Figure_10_01_202.jpg\" alt=\"\" \/><\/p>\n<p>35.&nbsp;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]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182751\/CNX_Precalc_Figure_10_01_204.jpg\" alt=\"\" \/><\/p>\n<p>37.&nbsp;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]<br \/>\nNote that this ellipse is a circle. The circle has only one focus, which coincides with the center.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182752\/CNX_Precalc_Figure_10_01_206.jpg\" alt=\"\" \/><\/p>\n<p>39.&nbsp;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]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182754\/CNX_Precalc_Figure_10_01_208.jpg\" alt=\"\" \/><\/p>\n<p>41.&nbsp;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]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182757\/CNX_Precalc_Figure_10_01_210.jpg\" alt=\"\" \/><\/p>\n<p>43.&nbsp;Center [latex]\\left(-2,1\\right)[\/latex], Vertices [latex]\\left(0,1\\right),\\left(-4,1\\right),\\left(-2,5\\right),\\left(-2,-3\\right)[\/latex], Foci [latex]\\left(-2,1+2\\sqrt{3}\\right),\\left(-2,1 - 2\\sqrt{3}\\right)[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182759\/CNX_Precalc_Figure_10_01_212.jpg\" alt=\"\" \/><\/p>\n<p>45.&nbsp;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]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27182801\/CNX_Precalc_Figure_10_01_214.jpg\" alt=\"\" \/><\/p>\n<p>47.&nbsp;[latex]\\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{29}=1[\/latex]<\/p>\n<p>49.&nbsp;[latex]\\frac{{\\left(x - 4\\right)}^{2}}{25}+\\frac{{\\left(y - 2\\right)}^{2}}{1}=1[\/latex]<\/p>\n<p>51.&nbsp;[latex]\\frac{{\\left(x+3\\right)}^{2}}{16}+\\frac{{\\left(y - 4\\right)}^{2}}{4}=1[\/latex]<\/p>\n<p>53.&nbsp;[latex]\\frac{{x}^{2}}{81}+\\frac{{y}^{2}}{9}=1[\/latex]<\/p>\n<p>55.&nbsp;[latex]\\frac{{\\left(x+2\\right)}^{2}}{4}+\\frac{{\\left(y - 2\\right)}^{2}}{9}=1[\/latex]<\/p>\n<p>57.&nbsp;[latex]\\text{Area}=12\\pi[\/latex] square units<\/p>\n<p>59.&nbsp;[latex]\\text{Area}=2\\sqrt{5}\\pi[\/latex] square units<\/p>\n<p>61.&nbsp;[latex]\\text{Area }9\\pi[\/latex] square units<\/p>\n<p>63.&nbsp;[latex]\\frac{{x}^{2}}{4{h}^{2}}+\\frac{{y}^{2}}{\\frac{1}{4}{h}^{2}}=1[\/latex]<\/p>\n<p>65.&nbsp;[latex]\\frac{{x}^{2}}{400}+\\frac{{y}^{2}}{144}=1[\/latex]. Distance = 17.32 feet<\/p>\n<p>67.&nbsp;Approximately 51.96 feet<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-1436\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":503070,"menu_order":7,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax College\",\"organization\":\"OpenStax\",\"url\":\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-1436","chapter","type-chapter","status-publish","hentry"],"part":1429,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1436","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/users\/503070"}],"version-history":[{"count":0,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1436\/revisions"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/parts\/1429"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1436\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/media?parent=1436"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapter-type?post=1436"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/contributor?post=1436"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/license?post=1436"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}