{"id":1442,"date":"2023-06-05T14:51:46","date_gmt":"2023-06-05T14:51:46","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-rotation-of-axes\/"},"modified":"2023-06-05T14:51:46","modified_gmt":"2023-06-05T14:51:46","slug":"solutions-for-rotation-of-axes","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/chapter\/solutions-for-rotation-of-axes\/","title":{"raw":"Solutions 68: Rotation of Axes","rendered":"Solutions 68: Rotation of Axes"},"content":{"raw":"\n<h2>Solutions to Odd-Numbered Exercises<\/h2>\n1.&nbsp;The [latex]xy[\/latex] term causes a rotation of the graph to occur.\n\n3.&nbsp;The conic section is a hyperbola.\n\n5.&nbsp;It gives the angle of rotation of the axes in order to eliminate the [latex]xy[\/latex] term.\n\n7.&nbsp;[latex]AB=0[\/latex], parabola\n\n9.&nbsp;[latex]AB=-4&lt;0[\/latex], hyperbola\n\n11.&nbsp;[latex]AB=6&gt;0[\/latex], ellipse\n\n13.&nbsp;[latex]{B}^{2}-4AC=0[\/latex], parabola\n\n15.&nbsp;[latex]{B}^{2}-4AC=0[\/latex], parabola\n\n17.&nbsp;[latex]{B}^{2}-4AC=-96&lt;0[\/latex], ellipse\n\n19.&nbsp;[latex]7{{x}^{\\prime }}^{2}+9{{y}^{\\prime }}^{2}-4=0[\/latex]\n\n21.&nbsp;[latex]3{{x}^{\\prime }}^{2}+2{x}^{\\prime }{y}^{\\prime }-5{{y}^{\\prime }}^{2}+1=0[\/latex]\n\n23.&nbsp;[latex]\\theta ={60}^{\\circ },11{{x}^{\\prime }}^{2}-{{y}^{\\prime }}^{2}+\\sqrt{3}{x}^{\\prime }+{y}^{\\prime }-4=0[\/latex]\n\n25.&nbsp;[latex]\\theta ={150}^{\\circ },21{{x}^{\\prime }}^{2}+9{{y}^{\\prime }}^{2}+4{x}^{\\prime }-4\\sqrt{3}{y}^{\\prime }-6=0[\/latex]\n\n27.&nbsp;[latex]\\theta \\approx {36.9}^{\\circ },125{{x}^{\\prime }}^{2}+6{x}^{\\prime }-42{y}^{\\prime }+10=0[\/latex]\n\n29.&nbsp;[latex]\\theta ={45}^{\\circ },3{{x}^{\\prime }}^{2}-{{y}^{\\prime }}^{2}-\\sqrt{2}{x}^{\\prime }+\\sqrt{2}{y}^{\\prime }+1=0[\/latex]\n\n31.&nbsp;[latex]\\frac{\\sqrt{2}}{2}\\left({x}^{\\prime }+{y}^{\\prime }\\right)=\\frac{1}{2}{\\left({x}^{\\prime }-{y}^{\\prime }\\right)}^{2}[\/latex]\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183052\/CNX_Precalc_Figure_10_04_2012.jpg\" alt=\"\">\n\n33.&nbsp;[latex]\\frac{{\\left({x}^{\\prime }-{y}^{\\prime }\\right)}^{2}}{8}+\\frac{{\\left({x}^{\\prime }+{y}^{\\prime }\\right)}^{2}}{2}=1[\/latex]\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183054\/CNX_Precalc_Figure_10_04_2032.jpg\" alt=\"\">\n\n35.&nbsp;[latex]\\frac{{\\left({x}^{\\prime }+{y}^{\\prime }\\right)}^{2}}{2}-\\frac{{\\left({x}^{\\prime }-{y}^{\\prime }\\right)}^{2}}{2}=1[\/latex]\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183056\/CNX_Precalc_Figure_10_04_2052.jpg\" alt=\"\">\n\n37.&nbsp;[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]\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183058\/CNX_Precalc_Figure_10_04_2072.jpg\" alt=\"\">\n\n39.\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183101\/CNX_Precalc_Figure_10_04_2092.jpg\" alt=\"\">\n\n41.\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183103\/CNX_Precalc_Figure_10_04_2112.jpg\" alt=\"\">\n\n43.\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183105\/CNX_Precalc_Figure_10_04_2132.jpg\" alt=\"\">\n\n45.\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183107\/CNX_Precalc_Figure_10_04_2152.jpg\" alt=\"\">\n\n47.\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183109\/CNX_Precalc_Figure_10_04_217.jpg\" alt=\"\">\n\n49.\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183111\/CNX_Precalc_Figure_10_04_219.jpg\" alt=\"\">\n\n51.&nbsp;[latex]\\theta ={45}^{\\circ }[\/latex]\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183113\/CNX_Precalc_Figure_10_04_221.jpg\" alt=\"\">\n\n53.&nbsp;[latex]\\theta ={60}^{\\circ }[\/latex]\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183115\/CNX_Precalc_Figure_10_04_223.jpg\" alt=\"\">\n\n55.&nbsp;[latex]\\theta \\approx {36.9}^{\\circ }[\/latex]\n<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183117\/CNX_Precalc_Figure_10_04_225.jpg\" alt=\"\">\n\n57. [latex]-4\\sqrt{6}&lt;k&lt;4\\sqrt{6}[\/latex]\n\n59.&nbsp;[latex]k=2[\/latex]\n","rendered":"<h2>Solutions to Odd-Numbered Exercises<\/h2>\n<p>1.&nbsp;The [latex]xy[\/latex] term causes a rotation of the graph to occur.<\/p>\n<p>3.&nbsp;The conic section is a hyperbola.<\/p>\n<p>5.&nbsp;It gives the angle of rotation of the axes in order to eliminate the [latex]xy[\/latex] term.<\/p>\n<p>7.&nbsp;[latex]AB=0[\/latex], parabola<\/p>\n<p>9.&nbsp;[latex]AB=-4<0[\/latex], hyperbola\n\n11.&nbsp;[latex]AB=6>0[\/latex], ellipse<\/p>\n<p>13.&nbsp;[latex]{B}^{2}-4AC=0[\/latex], parabola<\/p>\n<p>15.&nbsp;[latex]{B}^{2}-4AC=0[\/latex], parabola<\/p>\n<p>17.&nbsp;[latex]{B}^{2}-4AC=-96<0[\/latex], ellipse\n\n19.&nbsp;[latex]7{{x}^{\\prime }}^{2}+9{{y}^{\\prime }}^{2}-4=0[\/latex]\n\n21.&nbsp;[latex]3{{x}^{\\prime }}^{2}+2{x}^{\\prime }{y}^{\\prime }-5{{y}^{\\prime }}^{2}+1=0[\/latex]\n\n23.&nbsp;[latex]\\theta ={60}^{\\circ },11{{x}^{\\prime }}^{2}-{{y}^{\\prime }}^{2}+\\sqrt{3}{x}^{\\prime }+{y}^{\\prime }-4=0[\/latex]\n\n25.&nbsp;[latex]\\theta ={150}^{\\circ },21{{x}^{\\prime }}^{2}+9{{y}^{\\prime }}^{2}+4{x}^{\\prime }-4\\sqrt{3}{y}^{\\prime }-6=0[\/latex]\n\n27.&nbsp;[latex]\\theta \\approx {36.9}^{\\circ },125{{x}^{\\prime }}^{2}+6{x}^{\\prime }-42{y}^{\\prime }+10=0[\/latex]\n\n29.&nbsp;[latex]\\theta ={45}^{\\circ },3{{x}^{\\prime }}^{2}-{{y}^{\\prime }}^{2}-\\sqrt{2}{x}^{\\prime }+\\sqrt{2}{y}^{\\prime }+1=0[\/latex]\n\n31.&nbsp;[latex]\\frac{\\sqrt{2}}{2}\\left({x}^{\\prime }+{y}^{\\prime }\\right)=\\frac{1}{2}{\\left({x}^{\\prime }-{y}^{\\prime }\\right)}^{2}[\/latex]\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183052\/CNX_Precalc_Figure_10_04_2012.jpg\" alt=\"\" \/><\/p>\n<p>33.&nbsp;[latex]\\frac{{\\left({x}^{\\prime }-{y}^{\\prime }\\right)}^{2}}{8}+\\frac{{\\left({x}^{\\prime }+{y}^{\\prime }\\right)}^{2}}{2}=1[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183054\/CNX_Precalc_Figure_10_04_2032.jpg\" alt=\"\" \/><\/p>\n<p>35.&nbsp;[latex]\\frac{{\\left({x}^{\\prime }+{y}^{\\prime }\\right)}^{2}}{2}-\\frac{{\\left({x}^{\\prime }-{y}^{\\prime }\\right)}^{2}}{2}=1[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183056\/CNX_Precalc_Figure_10_04_2052.jpg\" alt=\"\" \/><\/p>\n<p>37.&nbsp;[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]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183058\/CNX_Precalc_Figure_10_04_2072.jpg\" alt=\"\" \/><\/p>\n<p>39.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183101\/CNX_Precalc_Figure_10_04_2092.jpg\" alt=\"\" \/><\/p>\n<p>41.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183103\/CNX_Precalc_Figure_10_04_2112.jpg\" alt=\"\" \/><\/p>\n<p>43.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183105\/CNX_Precalc_Figure_10_04_2132.jpg\" alt=\"\" \/><\/p>\n<p>45.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183107\/CNX_Precalc_Figure_10_04_2152.jpg\" alt=\"\" \/><\/p>\n<p>47.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183109\/CNX_Precalc_Figure_10_04_217.jpg\" alt=\"\" \/><\/p>\n<p>49.<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183111\/CNX_Precalc_Figure_10_04_219.jpg\" alt=\"\" \/><\/p>\n<p>51.&nbsp;[latex]\\theta ={45}^{\\circ }[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183113\/CNX_Precalc_Figure_10_04_221.jpg\" alt=\"\" \/><\/p>\n<p>53.&nbsp;[latex]\\theta ={60}^{\\circ }[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183115\/CNX_Precalc_Figure_10_04_223.jpg\" alt=\"\" \/><\/p>\n<p>55.&nbsp;[latex]\\theta \\approx {36.9}^{\\circ }[\/latex]<br \/>\n<img decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/3675\/2018\/09\/27183117\/CNX_Precalc_Figure_10_04_225.jpg\" alt=\"\" \/><\/p>\n<p>57. [latex]-4\\sqrt{6}<k<4\\sqrt{6}[\/latex]\n\n59.&nbsp;[latex]k=2[\/latex]\n\n<\/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-1442\">\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":13,"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-1442","chapter","type-chapter","status-publish","hentry"],"part":1429,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapters\/1442","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\/1442\/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\/1442\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/media?parent=1442"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/pressbooks\/v2\/chapter-type?post=1442"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/contributor?post=1442"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/tulsacc-math1613\/wp-json\/wp\/v2\/license?post=1442"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}