{"id":349,"date":"2026-02-16T20:57:43","date_gmt":"2026-02-16T20:57:43","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/?post_type=chapter&p=349"},"modified":"2026-02-18T20:36:56","modified_gmt":"2026-02-18T20:36:56","slug":"conics-get-stronger-answer-key","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/chapter\/conics-get-stronger-answer-key\/","title":{"raw":"Conics: Get Stronger Answer Key","rendered":"Conics: Get Stronger Answer Key"},"content":{"raw":"

The Ellipse<\/h1>\r\n1.\u00a0An 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.\r\n\r\n3.\u00a0This special case would be a circle.\r\n\r\n5.\u00a0It is symmetric about the x<\/em>-axis, y<\/em>-axis, and the origin.\r\n\r\n11. [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].\r\n\r\n13.\u00a0[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].\r\n\r\n15.\u00a0[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].\r\n\r\n17.\u00a0[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].\r\n\r\n19.\u00a0[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].\r\n\r\n23.\u00a0[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].\r\n\r\n27.\u00a0Foci [latex]\\left(-3,-1+\\sqrt{11}\\right),\\left(-3,-1-\\sqrt{11}\\right)[\/latex]\r\n\r\n29.\u00a0Focus [latex]\\left(0,0\\right)[\/latex]\r\n\r\n31.\u00a0Foci [latex]\\left(-10,30\\right),\\left(-10,-30\\right)[\/latex]\r\n\r\n33.\u00a0Center [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]\r\n\"\"\r\n\r\n35.\u00a0Center [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]\r\n\"\"\r\n\r\n37.\u00a0Center [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]\r\nNote that this ellipse is a circle. The circle has only one focus, which coincides with the center.\r\n\"\"\r\n\r\n39.\u00a0Center [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]\r\n\"\"\r\n\r\n41.\u00a0Center [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]\r\n\"\"\r\n\r\n45.\u00a0Center [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]\r\n\"\"\r\n\r\n47.\u00a0[latex]\\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{29}=1[\/latex]\r\n\r\n49.\u00a0[latex]\\frac{{\\left(x - 4\\right)}^{2}}{25}+\\frac{{\\left(y - 2\\right)}^{2}}{1}=1[\/latex]\r\n\r\n53.\u00a0[latex]\\frac{{x}^{2}}{81}+\\frac{{y}^{2}}{9}=1[\/latex]\r\n\r\n55.\u00a0[latex]\\frac{{\\left(x+2\\right)}^{2}}{4}+\\frac{{\\left(y - 2\\right)}^{2}}{9}=1[\/latex]\r\n\r\n63.\u00a0[latex]\\frac{{x}^{2}}{4{h}^{2}}+\\frac{{y}^{2}}{\\frac{1}{4}{h}^{2}}=1[\/latex]\r\n\r\n65.\u00a0[latex]\\frac{{x}^{2}}{400}+\\frac{{y}^{2}}{144}=1[\/latex]. Distance = 17.32 feet\r\n\r\n67.\u00a0Approximately 51.96 feet\r\n

The Hyperbola<\/h1>\r\n5.\u00a0The center must be the midpoint of the line segment joining the foci.\r\n\r\n11.\u00a0[latex]\\frac{{x}^{2}}{{5}^{2}}-\\frac{{y}^{2}}{{6}^{2}}=1[\/latex]; vertices: [latex]\\left(5,0\\right),\\left(-5,0\\right)[\/latex]; foci: [latex]\\left(\\sqrt{61},0\\right),\\left(-\\sqrt{61},0\\right)[\/latex]; asymptotes: [latex]y=\\frac{6}{5}x,y=-\\frac{6}{5}x[\/latex]\r\n\r\n13.\u00a0[latex]\\frac{{y}^{2}}{{2}^{2}}-\\frac{{x}^{2}}{{9}^{2}}=1[\/latex]; vertices: [latex]\\left(0,2\\right),\\left(0,-2\\right)[\/latex]; foci: [latex]\\left(0,\\sqrt{85}\\right),\\left(0,-\\sqrt{85}\\right)[\/latex]; asymptotes: [latex]y=\\frac{2}{9}x,y=-\\frac{2}{9}x[\/latex]\r\n\r\n15.\u00a0[latex]\\frac{{\\left(x - 1\\right)}^{2}}{{3}^{2}}-\\frac{{\\left(y - 2\\right)}^{2}}{{4}^{2}}=1[\/latex]; vertices: [latex]\\left(4,2\\right),\\left(-2,2\\right)[\/latex]; foci: [latex]\\left(6,2\\right),\\left(-4,2\\right)[\/latex]; asymptotes: [latex]y=\\frac{4}{3}\\left(x - 1\\right)+2,y=-\\frac{4}{3}\\left(x - 1\\right)+2[\/latex]\r\n\r\n17.\u00a0[latex]\\frac{{\\left(x - 2\\right)}^{2}}{{7}^{2}}-\\frac{{\\left(y+7\\right)}^{2}}{{7}^{2}}=1[\/latex]; vertices: [latex]\\left(9,-7\\right),\\left(-5,-7\\right)[\/latex]; foci: [latex]\\left(2+7\\sqrt{2},-7\\right),\\left(2 - 7\\sqrt{2},-7\\right)[\/latex]; asymptotes: [latex]y=x - 9,y=-x - 5[\/latex]\r\n\r\n19.\u00a0[latex]\\frac{{\\left(x+3\\right)}^{2}}{{3}^{2}}-\\frac{{\\left(y - 3\\right)}^{2}}{{3}^{2}}=1[\/latex]; vertices: [latex]\\left(0,3\\right),\\left(-6,3\\right)[\/latex]; foci: [latex]\\left(-3+3\\sqrt{2},1\\right),\\left(-3 - 3\\sqrt{2},1\\right)[\/latex]; asymptotes: [latex]y=x+6,y=-x[\/latex]\r\n\r\n23.\u00a0[latex]\\frac{{\\left(y+5\\right)}^{2}}{{7}^{2}}-\\frac{{\\left(x+1\\right)}^{2}}{{70}^{2}}=1[\/latex]; vertices: [latex]\\left(-1,2\\right),\\left(-1,-12\\right)[\/latex]; foci: [latex]\\left(-1,-5+7\\sqrt{101}\\right),\\left(-1,-5 - 7\\sqrt{101}\\right)[\/latex]; asymptotes: [latex]y=\\frac{1}{10}\\left(x+1\\right)-5,y=-\\frac{1}{10}\\left(x+1\\right)-5[\/latex]\r\n\r\n27.\u00a0[latex]y=\\frac{2}{5}\\left(x - 3\\right)-4,y=-\\frac{2}{5}\\left(x - 3\\right)-4[\/latex]\r\n\r\n29.\u00a0[latex]y=\\frac{3}{4}\\left(x - 1\\right)+1,y=-\\frac{3}{4}\\left(x - 1\\right)+1[\/latex]\r\n\r\n31.\r\n\"\"\r\n\r\n33.\r\n\"\"\r\n\r\n35.\r\n\"\"\r\n\r\n37.\r\n\"\"\r\n\r\n39.\r\n\"\"\r\n\r\n41.\r\n\"\"\r\n\r\n43.\r\n\"\"\r\n\r\n45.\u00a0[latex]\\frac{{x}^{2}}{9}-\\frac{{y}^{2}}{16}=1[\/latex]\r\n\r\n47.\u00a0[latex]\\frac{{\\left(x - 6\\right)}^{2}}{25}-\\frac{{\\left(y - 1\\right)}^{2}}{11}=1[\/latex]\r\n\r\n49.\u00a0[latex]\\frac{{\\left(x - 4\\right)}^{2}}{25}-\\frac{{\\left(y - 2\\right)}^{2}}{1}=1[\/latex]\r\n\r\n51.\u00a0[latex]\\frac{{y}^{2}}{16}-\\frac{{x}^{2}}{25}=1[\/latex]\r\n\r\n53.\u00a0[latex]\\frac{{y}^{2}}{9}-\\frac{{\\left(x+1\\right)}^{2}}{9}=1[\/latex]\r\n\r\n55.\u00a0[latex]\\frac{{\\left(x+3\\right)}^{2}}{25}-\\frac{{\\left(y+3\\right)}^{2}}{25}=1[\/latex]\r\n\r\n61.\u00a0[latex]\\frac{{x}^{2}}{25}-\\frac{{y}^{2}}{25}=1[\/latex]\r\n\"\"\r\n\r\n63.\u00a0[latex]\\frac{{x}^{2}}{100}-\\frac{{y}^{2}}{25}=1[\/latex]\r\n\"\"\r\n\r\n65.\u00a0[latex]\\frac{{x}^{2}}{400}-\\frac{{y}^{2}}{225}=1[\/latex]\r\n\"\"\r\n\r\n67.\u00a0[latex]\\frac{{\\left(x - 1\\right)}^{2}}{0.25}-\\frac{{y}^{2}}{0.75}=1[\/latex]\r\n\r\n69.\u00a0[latex]\\frac{{\\left(x - 3\\right)}^{2}}{4}-\\frac{{y}^{2}}{5}=1[\/latex]\r\n

The Parabola<\/h1>\r\n1.\u00a0A parabola is the set of points in the plane that lie equidistant from a fixed point, the focus, and a fixed line, the directrix.\r\n\r\n3.\u00a0The graph will open down.\r\n\r\n5.\u00a0The distance between the focus and directrix will increase.\r\n\r\n11.\u00a0[latex]{y}^{2}=\\frac{1}{8}x,V:\\left(0,0\\right);F:\\left(\\frac{1}{32},0\\right);d:x=-\\frac{1}{32}[\/latex]\r\n\r\n13.\u00a0[latex]{x}^{2}=-\\frac{1}{4}y,V:\\left(0,0\\right);F:\\left(0,-\\frac{1}{16}\\right);d:y=\\frac{1}{16}[\/latex]\r\n\r\n19.\u00a0[latex]{\\left(y - 4\\right)}^{2}=2\\left(x+3\\right),V:\\left(-3,4\\right);F:\\left(-\\frac{5}{2},4\\right);d:x=-\\frac{7}{2}[\/latex]\r\n\r\n21.\u00a0[latex]{\\left(x+4\\right)}^{2}=24\\left(y+1\\right),V:\\left(-4,-1\\right);F:\\left(-4,5\\right);d:y=-7[\/latex]\r\n\r\n23.\u00a0[latex]{\\left(y - 3\\right)}^{2}=-12\\left(x+1\\right),V:\\left(-1,3\\right);F:\\left(-4,3\\right);d:x=2[\/latex]\r\n\r\n25.\u00a0[latex]{\\left(x - 5\\right)}^{2}=\\frac{4}{5}\\left(y+3\\right),V:\\left(5,-3\\right);F:\\left(5,-\\frac{14}{5}\\right);d:y=-\\frac{16}{5}[\/latex]\r\n\r\n27.\u00a0[latex]{\\left(x - 2\\right)}^{2}=-2\\left(y - 5\\right),V:\\left(2,5\\right);F:\\left(2,\\frac{9}{2}\\right);d:y=\\frac{11}{2}[\/latex]\r\n\r\n29.\u00a0[latex]{\\left(y - 1\\right)}^{2}=\\frac{4}{3}\\left(x - 5\\right),V:\\left(5,1\\right);F:\\left(\\frac{16}{3},1\\right);d:x=\\frac{14}{3}[\/latex]\r\n\r\n31.\r\n\"\"\r\n\r\n33.\r\n\"\"\r\n\r\n35.\r\n\"\"\r\n\r\n37.\r\n\"\"\r\n\r\n39.\r\n\"\"\r\n\r\n41.\r\n\"\"\r\n\r\n43.\r\n\"\"\r\n\r\n45.\u00a0[latex]{x}^{2}=-16y[\/latex]\r\n\r\n47.\u00a0[latex]{\\left(y - 2\\right)}^{2}=4\\sqrt{2}\\left(x - 2\\right)[\/latex]\r\n\r\n49.\u00a0[latex]{\\left(y+\\sqrt{3}\\right)}^{2}=-4\\sqrt{2}\\left(x-\\sqrt{2}\\right)[\/latex]\r\n\r\n51.\u00a0[latex]{x}^{2}=y[\/latex]\r\n\r\n53.\u00a0[latex]{\\left(y - 2\\right)}^{2}=\\frac{1}{4}\\left(x+2\\right)[\/latex]\r\n\r\n55.\u00a0[latex]{\\left(y-\\sqrt{3}\\right)}^{2}=4\\sqrt{5}\\left(x+\\sqrt{2}\\right)[\/latex]\r\n\r\n61.\u00a0[latex]\\left(0,1\\right)[\/latex]\r\n\r\n63.\u00a0At the point 2.25 feet above the vertex.\r\n\r\n65.\u00a00.5625 feet\r\n\r\n67.\u00a0[latex]{x}^{2}=-125\\left(y - 20\\right)[\/latex],\u00a0height is 7.2 feet\r\n\r\n69.\u00a02304 feet\r\n

Rotation of Axes<\/h1>\r\n1.\u00a0The [latex]xy[\/latex] term causes a rotation of the graph to occur.\r\n\r\n3.\u00a0The conic section is a hyperbola.\r\n\r\n5.\u00a0It gives the angle of rotation of the axes in order to eliminate the [latex]xy[\/latex] term.\r\n\r\n7.\u00a0[latex]AB=0[\/latex], parabola\r\n\r\n9.\u00a0[latex]AB=-4<0[\/latex], hyperbola\r\n\r\n11.\u00a0[latex]AB=6>0[\/latex], ellipse\r\n\r\n13.\u00a0[latex]{B}^{2}-4AC=0[\/latex], parabola\r\n\r\n15.\u00a0[latex]{B}^{2}-4AC=0[\/latex], parabola\r\n\r\n17.\u00a0[latex]{B}^{2}-4AC=-96<0[\/latex], ellipse\r\n\r\n19.\u00a0[latex]7{{x}^{\\prime }}^{2}+9{{y}^{\\prime }}^{2}-4=0[\/latex]\r\n\r\n21.\u00a0[latex]3{{x}^{\\prime }}^{2}+2{x}^{\\prime }{y}^{\\prime }-5{{y}^{\\prime }}^{2}+1=0[\/latex]\r\n\r\n23.\u00a0[latex]\\theta ={60}^{\\circ },11{{x}^{\\prime }}^{2}-{{y}^{\\prime }}^{2}+\\sqrt{3}{x}^{\\prime }+{y}^{\\prime }-4=0[\/latex]\r\n\r\n25.\u00a0[latex]\\theta ={150}^{\\circ },21{{x}^{\\prime }}^{2}+9{{y}^{\\prime }}^{2}+4{x}^{\\prime }-4\\sqrt{3}{y}^{\\prime }-6=0[\/latex]\r\n\r\n27.\u00a0[latex]\\theta \\approx {36.9}^{\\circ },125{{x}^{\\prime }}^{2}+6{x}^{\\prime }-42{y}^{\\prime }+10=0[\/latex]\r\n\r\n29.\u00a0[latex]\\theta ={45}^{\\circ },3{{x}^{\\prime }}^{2}-{{y}^{\\prime }}^{2}-\\sqrt{2}{x}^{\\prime }+\\sqrt{2}{y}^{\\prime }+1=0[\/latex]\r\n\r\n31.\u00a0[latex]\\frac{\\sqrt{2}}{2}\\left({x}^{\\prime }+{y}^{\\prime }\\right)=\\frac{1}{2}{\\left({x}^{\\prime }-{y}^{\\prime }\\right)}^{2}[\/latex]\r\n\"\"\r\n\r\n33.\u00a0[latex]\\frac{{\\left({x}^{\\prime }-{y}^{\\prime }\\right)}^{2}}{8}+\\frac{{\\left({x}^{\\prime }+{y}^{\\prime }\\right)}^{2}}{2}=1[\/latex]\r\n\"\"\r\n\r\n35.\u00a0[latex]\\frac{{\\left({x}^{\\prime }+{y}^{\\prime }\\right)}^{2}}{2}-\\frac{{\\left({x}^{\\prime }-{y}^{\\prime }\\right)}^{2}}{2}=1[\/latex]\r\n\"\"\r\n\r\n37.\u00a0[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]\r\n\"\"\r\n\r\n39.\r\n\"\"\r\n\r\n41.\r\n\"\"\r\n\r\n43.\r\n\"\"\r\n\r\n45.\r\n\"\"\r\n\r\n47.\r\n\"\"\r\n\r\n49.\r\n\"\"\r\n\r\n51.\u00a0[latex]\\theta ={45}^{\\circ }[\/latex]\r\n\"\"\r\n\r\n53.\u00a0[latex]\\theta ={60}^{\\circ }[\/latex]\r\n\"\"\r\n\r\n55.\u00a0[latex]\\theta \\approx {36.9}^{\\circ }[\/latex]\r\n\"\"\r\n\r\n57. [latex]-4\\sqrt{6}<k<4\\sqrt{6}[\/latex]\r\n\r\n59.\u00a0[latex]k=2[\/latex]\r\n

Conic Sections in Polar Coordinates<\/h1>\r\n1.\u00a0If eccentricity is less than 1, it is an ellipse. If eccentricity is equal to 1, it is a parabola. If eccentricity is greater than 1, it is a hyperbola.\r\n\r\n7.\u00a0Parabola with [latex]e=1[\/latex] and directrix [latex]\\frac{3}{4}[\/latex] units below the pole.\r\n\r\n9.\u00a0Hyperbola with [latex]e=2[\/latex] and directrix [latex]\\frac{5}{2}[\/latex] units above the pole.\r\n\r\n11.\u00a0Parabola with [latex]e=1[\/latex] and directrix [latex]\\frac{3}{10}[\/latex] units to the right of the pole.\r\n\r\n13.\u00a0Ellipse with [latex]e=\\frac{2}{7}[\/latex] and directrix [latex]2[\/latex] units to the right of the pole.\r\n\r\n15.\u00a0Hyperbola with [latex]e=\\frac{5}{3}[\/latex] and directrix [latex]\\frac{11}{5}[\/latex] units above the pole.\r\n\r\n17.\u00a0Hyperbola with [latex]e=\\frac{8}{7}[\/latex] and directrix [latex]\\frac{7}{8}[\/latex] units to the right of the pole.\r\n\r\n19.\u00a0[latex]25{x}^{2}+16{y}^{2}-12y - 4=0[\/latex]\r\n\r\n21.\u00a0[latex]21{x}^{2}-4{y}^{2}-30x+9=0[\/latex]\r\n\r\n23.\u00a0[latex]64{y}^{2}=48x+9[\/latex]\r\n\r\n25.\u00a0[latex]96{y}^{2}-25{x}^{2}+110y+25=0[\/latex]\r\n\r\n27.\u00a0[latex]3{x}^{2}+4{y}^{2}-2x - 1=0[\/latex]\r\n\r\n29.\u00a0[latex]5{x}^{2}+9{y}^{2}-24x - 36=0[\/latex]\r\n\r\n31.\r\n\"\"\r\n\r\n33.\r\n\"\"\r\n\r\n35.\r\n\"\"\r\n\r\n37.\r\n\"\"\r\n\r\n39.\r\n\"\"\r\n\r\n41.\r\n\"\"\r\n\r\n43.\u00a0[latex]r=\\frac{4}{5+\\cos \\theta }[\/latex]\r\n\r\n45.\u00a0[latex]r=\\frac{4}{1+2\\sin \\theta }[\/latex]\r\n\r\n47.\u00a0[latex]r=\\frac{1}{1+\\cos \\theta }[\/latex]\r\n\r\n49.\u00a0[latex]r=\\frac{7}{8 - 28\\cos \\theta }[\/latex]\r\n\r\n51.\u00a0[latex]r=\\frac{12}{2+3\\sin \\theta }[\/latex]\r\n\r\n53.\u00a0[latex]r=\\frac{15}{4 - 3\\cos \\theta }[\/latex]\r\n\r\n55.\u00a0[latex]r=\\frac{3}{3 - 3\\cos \\theta }[\/latex]\r\n\r\n57.\u00a0[latex]r=\\pm \\frac{2}{\\sqrt{1+\\sin \\theta \\cos \\theta }}[\/latex]\r\n\r\n59.\u00a0[latex]r=\\pm \\frac{2}{4\\cos \\theta +3\\sin \\theta }[\/latex]","rendered":"

The Ellipse<\/h1>\n

1.\u00a0An 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

3.\u00a0This special case would be a circle.<\/p>\n

5.\u00a0It is symmetric about the x<\/em>-axis, y<\/em>-axis, and the origin.<\/p>\n

11. [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

13.\u00a0[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

15.\u00a0[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

17.\u00a0[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

19.\u00a0[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

23.\u00a0[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

27.\u00a0Foci [latex]\\left(-3,-1+\\sqrt{11}\\right),\\left(-3,-1-\\sqrt{11}\\right)[\/latex]<\/p>\n

29.\u00a0Focus [latex]\\left(0,0\\right)[\/latex]<\/p>\n

31.\u00a0Foci [latex]\\left(-10,30\\right),\\left(-10,-30\\right)[\/latex]<\/p>\n

33.\u00a0Center [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\"\"<\/p>\n

35.\u00a0Center [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\"\"<\/p>\n

37.\u00a0Center [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\"\"<\/p>\n

39.\u00a0Center [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\"\"<\/p>\n

41.\u00a0Center [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\"\"<\/p>\n

45.\u00a0Center [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\"\"<\/p>\n

47.\u00a0[latex]\\frac{{x}^{2}}{25}+\\frac{{y}^{2}}{29}=1[\/latex]<\/p>\n

49.\u00a0[latex]\\frac{{\\left(x - 4\\right)}^{2}}{25}+\\frac{{\\left(y - 2\\right)}^{2}}{1}=1[\/latex]<\/p>\n

53.\u00a0[latex]\\frac{{x}^{2}}{81}+\\frac{{y}^{2}}{9}=1[\/latex]<\/p>\n

55.\u00a0[latex]\\frac{{\\left(x+2\\right)}^{2}}{4}+\\frac{{\\left(y - 2\\right)}^{2}}{9}=1[\/latex]<\/p>\n

63.\u00a0[latex]\\frac{{x}^{2}}{4{h}^{2}}+\\frac{{y}^{2}}{\\frac{1}{4}{h}^{2}}=1[\/latex]<\/p>\n

65.\u00a0[latex]\\frac{{x}^{2}}{400}+\\frac{{y}^{2}}{144}=1[\/latex]. Distance = 17.32 feet<\/p>\n

67.\u00a0Approximately 51.96 feet<\/p>\n

The Hyperbola<\/h1>\n

5.\u00a0The center must be the midpoint of the line segment joining the foci.<\/p>\n

11.\u00a0[latex]\\frac{{x}^{2}}{{5}^{2}}-\\frac{{y}^{2}}{{6}^{2}}=1[\/latex]; vertices: [latex]\\left(5,0\\right),\\left(-5,0\\right)[\/latex]; foci: [latex]\\left(\\sqrt{61},0\\right),\\left(-\\sqrt{61},0\\right)[\/latex]; asymptotes: [latex]y=\\frac{6}{5}x,y=-\\frac{6}{5}x[\/latex]<\/p>\n

13.\u00a0[latex]\\frac{{y}^{2}}{{2}^{2}}-\\frac{{x}^{2}}{{9}^{2}}=1[\/latex]; vertices: [latex]\\left(0,2\\right),\\left(0,-2\\right)[\/latex]; foci: [latex]\\left(0,\\sqrt{85}\\right),\\left(0,-\\sqrt{85}\\right)[\/latex]; asymptotes: [latex]y=\\frac{2}{9}x,y=-\\frac{2}{9}x[\/latex]<\/p>\n

15.\u00a0[latex]\\frac{{\\left(x - 1\\right)}^{2}}{{3}^{2}}-\\frac{{\\left(y - 2\\right)}^{2}}{{4}^{2}}=1[\/latex]; vertices: [latex]\\left(4,2\\right),\\left(-2,2\\right)[\/latex]; foci: [latex]\\left(6,2\\right),\\left(-4,2\\right)[\/latex]; asymptotes: [latex]y=\\frac{4}{3}\\left(x - 1\\right)+2,y=-\\frac{4}{3}\\left(x - 1\\right)+2[\/latex]<\/p>\n

17.\u00a0[latex]\\frac{{\\left(x - 2\\right)}^{2}}{{7}^{2}}-\\frac{{\\left(y+7\\right)}^{2}}{{7}^{2}}=1[\/latex]; vertices: [latex]\\left(9,-7\\right),\\left(-5,-7\\right)[\/latex]; foci: [latex]\\left(2+7\\sqrt{2},-7\\right),\\left(2 - 7\\sqrt{2},-7\\right)[\/latex]; asymptotes: [latex]y=x - 9,y=-x - 5[\/latex]<\/p>\n

19.\u00a0[latex]\\frac{{\\left(x+3\\right)}^{2}}{{3}^{2}}-\\frac{{\\left(y - 3\\right)}^{2}}{{3}^{2}}=1[\/latex]; vertices: [latex]\\left(0,3\\right),\\left(-6,3\\right)[\/latex]; foci: [latex]\\left(-3+3\\sqrt{2},1\\right),\\left(-3 - 3\\sqrt{2},1\\right)[\/latex]; asymptotes: [latex]y=x+6,y=-x[\/latex]<\/p>\n

23.\u00a0[latex]\\frac{{\\left(y+5\\right)}^{2}}{{7}^{2}}-\\frac{{\\left(x+1\\right)}^{2}}{{70}^{2}}=1[\/latex]; vertices: [latex]\\left(-1,2\\right),\\left(-1,-12\\right)[\/latex]; foci: [latex]\\left(-1,-5+7\\sqrt{101}\\right),\\left(-1,-5 - 7\\sqrt{101}\\right)[\/latex]; asymptotes: [latex]y=\\frac{1}{10}\\left(x+1\\right)-5,y=-\\frac{1}{10}\\left(x+1\\right)-5[\/latex]<\/p>\n

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

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

31.
\n\"\"<\/p>\n

33.
\n\"\"<\/p>\n

35.
\n\"\"<\/p>\n

37.
\n\"\"<\/p>\n

39.
\n\"\"<\/p>\n

41.
\n\"\"<\/p>\n

43.
\n\"\"<\/p>\n

45.\u00a0[latex]\\frac{{x}^{2}}{9}-\\frac{{y}^{2}}{16}=1[\/latex]<\/p>\n

47.\u00a0[latex]\\frac{{\\left(x - 6\\right)}^{2}}{25}-\\frac{{\\left(y - 1\\right)}^{2}}{11}=1[\/latex]<\/p>\n

49.\u00a0[latex]\\frac{{\\left(x - 4\\right)}^{2}}{25}-\\frac{{\\left(y - 2\\right)}^{2}}{1}=1[\/latex]<\/p>\n

51.\u00a0[latex]\\frac{{y}^{2}}{16}-\\frac{{x}^{2}}{25}=1[\/latex]<\/p>\n

53.\u00a0[latex]\\frac{{y}^{2}}{9}-\\frac{{\\left(x+1\\right)}^{2}}{9}=1[\/latex]<\/p>\n

55.\u00a0[latex]\\frac{{\\left(x+3\\right)}^{2}}{25}-\\frac{{\\left(y+3\\right)}^{2}}{25}=1[\/latex]<\/p>\n

61.\u00a0[latex]\\frac{{x}^{2}}{25}-\\frac{{y}^{2}}{25}=1[\/latex]
\n\"\"<\/p>\n

63.\u00a0[latex]\\frac{{x}^{2}}{100}-\\frac{{y}^{2}}{25}=1[\/latex]
\n\"\"<\/p>\n

65.\u00a0[latex]\\frac{{x}^{2}}{400}-\\frac{{y}^{2}}{225}=1[\/latex]
\n\"\"<\/p>\n

67.\u00a0[latex]\\frac{{\\left(x - 1\\right)}^{2}}{0.25}-\\frac{{y}^{2}}{0.75}=1[\/latex]<\/p>\n

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

The Parabola<\/h1>\n

1.\u00a0A parabola is the set of points in the plane that lie equidistant from a fixed point, the focus, and a fixed line, the directrix.<\/p>\n

3.\u00a0The graph will open down.<\/p>\n

5.\u00a0The distance between the focus and directrix will increase.<\/p>\n

11.\u00a0[latex]{y}^{2}=\\frac{1}{8}x,V:\\left(0,0\\right);F:\\left(\\frac{1}{32},0\\right);d:x=-\\frac{1}{32}[\/latex]<\/p>\n

13.\u00a0[latex]{x}^{2}=-\\frac{1}{4}y,V:\\left(0,0\\right);F:\\left(0,-\\frac{1}{16}\\right);d:y=\\frac{1}{16}[\/latex]<\/p>\n

19.\u00a0[latex]{\\left(y - 4\\right)}^{2}=2\\left(x+3\\right),V:\\left(-3,4\\right);F:\\left(-\\frac{5}{2},4\\right);d:x=-\\frac{7}{2}[\/latex]<\/p>\n

21.\u00a0[latex]{\\left(x+4\\right)}^{2}=24\\left(y+1\\right),V:\\left(-4,-1\\right);F:\\left(-4,5\\right);d:y=-7[\/latex]<\/p>\n

23.\u00a0[latex]{\\left(y - 3\\right)}^{2}=-12\\left(x+1\\right),V:\\left(-1,3\\right);F:\\left(-4,3\\right);d:x=2[\/latex]<\/p>\n

25.\u00a0[latex]{\\left(x - 5\\right)}^{2}=\\frac{4}{5}\\left(y+3\\right),V:\\left(5,-3\\right);F:\\left(5,-\\frac{14}{5}\\right);d:y=-\\frac{16}{5}[\/latex]<\/p>\n

27.\u00a0[latex]{\\left(x - 2\\right)}^{2}=-2\\left(y - 5\\right),V:\\left(2,5\\right);F:\\left(2,\\frac{9}{2}\\right);d:y=\\frac{11}{2}[\/latex]<\/p>\n

29.\u00a0[latex]{\\left(y - 1\\right)}^{2}=\\frac{4}{3}\\left(x - 5\\right),V:\\left(5,1\\right);F:\\left(\\frac{16}{3},1\\right);d:x=\\frac{14}{3}[\/latex]<\/p>\n

31.
\n\"\"<\/p>\n

33.
\n\"\"<\/p>\n

35.
\n\"\"<\/p>\n

37.
\n\"\"<\/p>\n

39.
\n\"\"<\/p>\n

41.
\n\"\"<\/p>\n

43.
\n\"\"<\/p>\n

45.\u00a0[latex]{x}^{2}=-16y[\/latex]<\/p>\n

47.\u00a0[latex]{\\left(y - 2\\right)}^{2}=4\\sqrt{2}\\left(x - 2\\right)[\/latex]<\/p>\n

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

51.\u00a0[latex]{x}^{2}=y[\/latex]<\/p>\n

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

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

61.\u00a0[latex]\\left(0,1\\right)[\/latex]<\/p>\n

63.\u00a0At the point 2.25 feet above the vertex.<\/p>\n

65.\u00a00.5625 feet<\/p>\n

67.\u00a0[latex]{x}^{2}=-125\\left(y - 20\\right)[\/latex],\u00a0height is 7.2 feet<\/p>\n

69.\u00a02304 feet<\/p>\n

Rotation of Axes<\/h1>\n

1.\u00a0The [latex]xy[\/latex] term causes a rotation of the graph to occur.<\/p>\n

3.\u00a0The conic section is a hyperbola.<\/p>\n

5.\u00a0It gives the angle of rotation of the axes in order to eliminate the [latex]xy[\/latex] term.<\/p>\n

7.\u00a0[latex]AB=0[\/latex], parabola<\/p>\n

9.\u00a0[latex]AB=-4<0[\/latex], hyperbola\n\n11.\u00a0[latex]AB=6>0[\/latex], ellipse<\/p>\n

13.\u00a0[latex]{B}^{2}-4AC=0[\/latex], parabola<\/p>\n

15.\u00a0[latex]{B}^{2}-4AC=0[\/latex], parabola<\/p>\n

17.\u00a0[latex]{B}^{2}-4AC=-96<0[\/latex], ellipse\n\n19.\u00a0[latex]7{{x}^{\\prime }}^{2}+9{{y}^{\\prime }}^{2}-4=0[\/latex]\n\n21.\u00a0[latex]3{{x}^{\\prime }}^{2}+2{x}^{\\prime }{y}^{\\prime }-5{{y}^{\\prime }}^{2}+1=0[\/latex]\n\n23.\u00a0[latex]\\theta ={60}^{\\circ },11{{x}^{\\prime }}^{2}-{{y}^{\\prime }}^{2}+\\sqrt{3}{x}^{\\prime }+{y}^{\\prime }-4=0[\/latex]\n\n25.\u00a0[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.\u00a0[latex]\\theta \\approx {36.9}^{\\circ },125{{x}^{\\prime }}^{2}+6{x}^{\\prime }-42{y}^{\\prime }+10=0[\/latex]\n\n29.\u00a0[latex]\\theta ={45}^{\\circ },3{{x}^{\\prime }}^{2}-{{y}^{\\prime }}^{2}-\\sqrt{2}{x}^{\\prime }+\\sqrt{2}{y}^{\\prime }+1=0[\/latex]\n\n31.\u00a0[latex]\\frac{\\sqrt{2}}{2}\\left({x}^{\\prime }+{y}^{\\prime }\\right)=\\frac{1}{2}{\\left({x}^{\\prime }-{y}^{\\prime }\\right)}^{2}[\/latex]\n\"\"<\/p>\n

33.\u00a0[latex]\\frac{{\\left({x}^{\\prime }-{y}^{\\prime }\\right)}^{2}}{8}+\\frac{{\\left({x}^{\\prime }+{y}^{\\prime }\\right)}^{2}}{2}=1[\/latex]
\n\"\"<\/p>\n

35.\u00a0[latex]\\frac{{\\left({x}^{\\prime }+{y}^{\\prime }\\right)}^{2}}{2}-\\frac{{\\left({x}^{\\prime }-{y}^{\\prime }\\right)}^{2}}{2}=1[\/latex]
\n\"\"<\/p>\n

37.\u00a0[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\"\"<\/p>\n

39.
\n\"\"<\/p>\n

41.
\n\"\"<\/p>\n

43.
\n\"\"<\/p>\n

45.
\n\"\"<\/p>\n

47.
\n\"\"<\/p>\n

49.
\n\"\"<\/p>\n

51.\u00a0[latex]\\theta ={45}^{\\circ }[\/latex]
\n\"\"<\/p>\n

53.\u00a0[latex]\\theta ={60}^{\\circ }[\/latex]
\n\"\"<\/p>\n

55.\u00a0[latex]\\theta \\approx {36.9}^{\\circ }[\/latex]
\n\"\"<\/p>\n

57. [latex]-4\\sqrt{6}Conic Sections in Polar Coordinates<\/h1>\n

1.\u00a0If eccentricity is less than 1, it is an ellipse. If eccentricity is equal to 1, it is a parabola. If eccentricity is greater than 1, it is a hyperbola.<\/p>\n

7.\u00a0Parabola with [latex]e=1[\/latex] and directrix [latex]\\frac{3}{4}[\/latex] units below the pole.<\/p>\n

9.\u00a0Hyperbola with [latex]e=2[\/latex] and directrix [latex]\\frac{5}{2}[\/latex] units above the pole.<\/p>\n

11.\u00a0Parabola with [latex]e=1[\/latex] and directrix [latex]\\frac{3}{10}[\/latex] units to the right of the pole.<\/p>\n

13.\u00a0Ellipse with [latex]e=\\frac{2}{7}[\/latex] and directrix [latex]2[\/latex] units to the right of the pole.<\/p>\n

15.\u00a0Hyperbola with [latex]e=\\frac{5}{3}[\/latex] and directrix [latex]\\frac{11}{5}[\/latex] units above the pole.<\/p>\n

17.\u00a0Hyperbola with [latex]e=\\frac{8}{7}[\/latex] and directrix [latex]\\frac{7}{8}[\/latex] units to the right of the pole.<\/p>\n

19.\u00a0[latex]25{x}^{2}+16{y}^{2}-12y - 4=0[\/latex]<\/p>\n

21.\u00a0[latex]21{x}^{2}-4{y}^{2}-30x+9=0[\/latex]<\/p>\n

23.\u00a0[latex]64{y}^{2}=48x+9[\/latex]<\/p>\n

25.\u00a0[latex]96{y}^{2}-25{x}^{2}+110y+25=0[\/latex]<\/p>\n

27.\u00a0[latex]3{x}^{2}+4{y}^{2}-2x - 1=0[\/latex]<\/p>\n

29.\u00a0[latex]5{x}^{2}+9{y}^{2}-24x - 36=0[\/latex]<\/p>\n

31.
\n\"\"<\/p>\n

33.
\n\"\"<\/p>\n

35.
\n\"\"<\/p>\n

37.
\n\"\"<\/p>\n

39.
\n\"\"<\/p>\n

41.
\n\"\"<\/p>\n

43.\u00a0[latex]r=\\frac{4}{5+\\cos \\theta }[\/latex]<\/p>\n

45.\u00a0[latex]r=\\frac{4}{1+2\\sin \\theta }[\/latex]<\/p>\n

47.\u00a0[latex]r=\\frac{1}{1+\\cos \\theta }[\/latex]<\/p>\n

49.\u00a0[latex]r=\\frac{7}{8 - 28\\cos \\theta }[\/latex]<\/p>\n

51.\u00a0[latex]r=\\frac{12}{2+3\\sin \\theta }[\/latex]<\/p>\n

53.\u00a0[latex]r=\\frac{15}{4 - 3\\cos \\theta }[\/latex]<\/p>\n

55.\u00a0[latex]r=\\frac{3}{3 - 3\\cos \\theta }[\/latex]<\/p>\n

57.\u00a0[latex]r=\\pm \\frac{2}{\\sqrt{1+\\sin \\theta \\cos \\theta }}[\/latex]<\/p>\n

59.\u00a0[latex]r=\\pm \\frac{2}{4\\cos \\theta +3\\sin \\theta }[\/latex]<\/p>\n","protected":false},"author":13,"menu_order":19,"template":"","meta":{"_candela_citation":"[]","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"part":224,"module-header":"- Select Header -","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\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/349"}],"collection":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/users\/13"}],"version-history":[{"count":3,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/349\/revisions"}],"predecessor-version":[{"id":371,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/349\/revisions\/371"}],"part":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/parts\/224"}],"metadata":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapters\/349\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/media?parent=349"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/pressbooks\/v2\/chapter-type?post=349"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/contributor?post=349"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/content.one.lumenlearning.com\/qrpracticepages\/wp-json\/wp\/v2\/license?post=349"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}