{"id":1903,"date":"2015-11-12T18:30:43","date_gmt":"2015-11-12T18:30:43","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1903"},"modified":"2015-11-12T18:30:43","modified_gmt":"2015-11-12T18:30:43","slug":"solutions-19","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/solutions-19\/","title":{"raw":"Solutions","rendered":"Solutions"},"content":{"raw":"

Solutions to Try Its<\/h2>\n1.\u00a0Vertices: [latex]\\left(\\pm 3,0\\right)[\/latex]; Foci: [latex]\\left(\\pm \\sqrt{34},0\\right)[\/latex]\n\n2.\u00a0[latex]\\frac{{y}^{2}}{4}-\\frac{{x}^{2}}{16}=1[\/latex]\n\n3.\u00a0[latex]\\frac{{\\left(y - 3\\right)}^{2}}{25}+\\frac{{\\left(x - 1\\right)}^{2}}{144}=1[\/latex]\n\n4.\u00a0vertices: [latex]\\left(\\pm 12,0\\right)[\/latex]; co-vertices: [latex]\\left(0,\\pm 9\\right)[\/latex]; foci: [latex]\\left(\\pm 15,0\\right)[\/latex]; asymptotes: [latex]y=\\pm \\frac{3}{4}x[\/latex];\n\"\"\n\n5.\u00a0center: [latex]\\left(3,-4\\right)[\/latex]; vertices: [latex]\\left(3,-14\\right)[\/latex] and [latex]\\left(3,6\\right)[\/latex]; co-vertices: [latex]\\left(-5,-4\\right)[\/latex]; and [latex]\\left(11,-4\\right)[\/latex]; foci: [latex]\\left(3,-4 - 2\\sqrt{41}\\right)[\/latex] and [latex]\\left(3,-4+2\\sqrt{41}\\right)[\/latex]; asymptotes: [latex]y=\\pm \\frac{5}{4}\\left(x - 3\\right)-4[\/latex]\n\"\"\n\n6.\u00a0The sides of the tower can be modeled by the hyperbolic equation. [latex]\\frac{{x}^{2}}{400}-\\frac{{y}^{2}}{3600}=1\\text{or }\\frac{{x}^{2}}{{20}^{2}}-\\frac{{y}^{2}}{{60}^{2}}=1[\/latex].\n

Solutions to Odd-Numbered Exercises<\/h2>\n1.\u00a0A hyperbola is the set of points in a plane the difference of whose distances from two fixed points (foci) is a positive constant.\n\n3.\u00a0The foci must lie on the transverse axis and be in the interior of the hyperbola.\n\n5.\u00a0The center must be the midpoint of the line segment joining the foci.\n\n7.\u00a0yes [latex]\\frac{{x}^{2}}{{6}^{2}}-\\frac{{y}^{2}}{{3}^{2}}=1[\/latex]\n\n9.\u00a0yes [latex]\\frac{{x}^{2}}{{4}^{2}}-\\frac{{y}^{2}}{{5}^{2}}=1[\/latex]\n\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]\n\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]\n\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]\n\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]\n\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]\n\n21.\u00a0[latex]\\frac{{\\left(y - 4\\right)}^{2}}{{2}^{2}}-\\frac{{\\left(x - 3\\right)}^{2}}{{4}^{2}}=1[\/latex]; vertices: [latex]\\left(3,6\\right),\\left(3,2\\right)[\/latex]; foci: [latex]\\left(3,4+2\\sqrt{5}\\right),\\left(3,4 - 2\\sqrt{5}\\right)[\/latex]; asymptotes: [latex]y=\\frac{1}{2}\\left(x - 3\\right)+4,y=-\\frac{1}{2}\\left(x - 3\\right)+4[\/latex]\n\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]\n\n25.\u00a0[latex]\\frac{{\\left(x+3\\right)}^{2}}{{5}^{2}}-\\frac{{\\left(y - 4\\right)}^{2}}{{2}^{2}}=1[\/latex]; vertices: [latex]\\left(2,4\\right),\\left(-8,4\\right)[\/latex]; foci: [latex]\\left(-3+\\sqrt{29},4\\right),\\left(-3-\\sqrt{29},4\\right)[\/latex]; asymptotes: [latex]y=\\frac{2}{5}\\left(x+3\\right)+4,y=-\\frac{2}{5}\\left(x+3\\right)+4[\/latex]\n\n27.\u00a0[latex]y=\\frac{2}{5}\\left(x - 3\\right)-4,y=-\\frac{2}{5}\\left(x - 3\\right)-4[\/latex]\n\n29.\u00a0[latex]y=\\frac{3}{4}\\left(x - 1\\right)+1,y=-\\frac{3}{4}\\left(x - 1\\right)+1[\/latex]\n\n31.\n\"\"\n\n33.\n\"\"\n\n35.\n\"\"\n\n37.\n\"\"\n\n39.\n\"\"\n\n41.\n\"\"\n\n43.\n\"\"\n\n45.\u00a0[latex]\\frac{{x}^{2}}{9}-\\frac{{y}^{2}}{16}=1[\/latex]\n\n47.\u00a0[latex]\\frac{{\\left(x - 6\\right)}^{2}}{25}-\\frac{{\\left(y - 1\\right)}^{2}}{11}=1[\/latex]\n\n49.\u00a0[latex]\\frac{{\\left(x - 4\\right)}^{2}}{25}-\\frac{{\\left(y - 2\\right)}^{2}}{1}=1[\/latex]\n\n51.\u00a0[latex]\\frac{{y}^{2}}{16}-\\frac{{x}^{2}}{25}=1[\/latex]\n\n53.\u00a0[latex]\\frac{{y}^{2}}{9}-\\frac{{\\left(x+1\\right)}^{2}}{9}=1[\/latex]\n\n55.\u00a0[latex]\\frac{{\\left(x+3\\right)}^{2}}{25}-\\frac{{\\left(y+3\\right)}^{2}}{25}=1[\/latex]\n\n57.\u00a0[latex]y\\left(x\\right)=3\\sqrt{{x}^{2}+1},y\\left(x\\right)=-3\\sqrt{{x}^{2}+1}[\/latex]\n\"\"\n\n59.\u00a0[latex]y\\left(x\\right)=1+2\\sqrt{{x}^{2}+4x+5},y\\left(x\\right)=1 - 2\\sqrt{{x}^{2}+4x+5}[\/latex]\n\"\"\n\n61.\u00a0[latex]\\frac{{x}^{2}}{25}-\\frac{{y}^{2}}{25}=1[\/latex]\n\"\"\n\n63.\u00a0[latex]\\frac{{x}^{2}}{100}-\\frac{{y}^{2}}{25}=1[\/latex]\n\"\"\n\n65.\u00a0[latex]\\frac{{x}^{2}}{400}-\\frac{{y}^{2}}{225}=1[\/latex]\n\"\"\n\n67.\u00a0[latex]\\frac{{\\left(x - 1\\right)}^{2}}{0.25}-\\frac{{y}^{2}}{0.75}=1[\/latex]\n\n69.\u00a0[latex]\\frac{{\\left(x - 3\\right)}^{2}}{4}-\\frac{{y}^{2}}{5}=1[\/latex]","rendered":"

Solutions to Try Its<\/h2>\n

1.\u00a0Vertices: [latex]\\left(\\pm 3,0\\right)[\/latex]; Foci: [latex]\\left(\\pm \\sqrt{34},0\\right)[\/latex]<\/p>\n

2.\u00a0[latex]\\frac{{y}^{2}}{4}-\\frac{{x}^{2}}{16}=1[\/latex]<\/p>\n

3.\u00a0[latex]\\frac{{\\left(y - 3\\right)}^{2}}{25}+\\frac{{\\left(x - 1\\right)}^{2}}{144}=1[\/latex]<\/p>\n

4.\u00a0vertices: [latex]\\left(\\pm 12,0\\right)[\/latex]; co-vertices: [latex]\\left(0,\\pm 9\\right)[\/latex]; foci: [latex]\\left(\\pm 15,0\\right)[\/latex]; asymptotes: [latex]y=\\pm \\frac{3}{4}x[\/latex];
\n\"\"<\/p>\n

5.\u00a0center: [latex]\\left(3,-4\\right)[\/latex]; vertices: [latex]\\left(3,-14\\right)[\/latex] and [latex]\\left(3,6\\right)[\/latex]; co-vertices: [latex]\\left(-5,-4\\right)[\/latex]; and [latex]\\left(11,-4\\right)[\/latex]; foci: [latex]\\left(3,-4 - 2\\sqrt{41}\\right)[\/latex] and [latex]\\left(3,-4+2\\sqrt{41}\\right)[\/latex]; asymptotes: [latex]y=\\pm \\frac{5}{4}\\left(x - 3\\right)-4[\/latex]
\n\"\"<\/p>\n

6.\u00a0The sides of the tower can be modeled by the hyperbolic equation. [latex]\\frac{{x}^{2}}{400}-\\frac{{y}^{2}}{3600}=1\\text{or }\\frac{{x}^{2}}{{20}^{2}}-\\frac{{y}^{2}}{{60}^{2}}=1[\/latex].<\/p>\n

Solutions to Odd-Numbered Exercises<\/h2>\n

1.\u00a0A hyperbola is the set of points in a plane the difference of whose distances from two fixed points (foci) is a positive constant.<\/p>\n

3.\u00a0The foci must lie on the transverse axis and be in the interior of the hyperbola.<\/p>\n

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

7.\u00a0yes [latex]\\frac{{x}^{2}}{{6}^{2}}-\\frac{{y}^{2}}{{3}^{2}}=1[\/latex]<\/p>\n

9.\u00a0yes [latex]\\frac{{x}^{2}}{{4}^{2}}-\\frac{{y}^{2}}{{5}^{2}}=1[\/latex]<\/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

21.\u00a0[latex]\\frac{{\\left(y - 4\\right)}^{2}}{{2}^{2}}-\\frac{{\\left(x - 3\\right)}^{2}}{{4}^{2}}=1[\/latex]; vertices: [latex]\\left(3,6\\right),\\left(3,2\\right)[\/latex]; foci: [latex]\\left(3,4+2\\sqrt{5}\\right),\\left(3,4 - 2\\sqrt{5}\\right)[\/latex]; asymptotes: [latex]y=\\frac{1}{2}\\left(x - 3\\right)+4,y=-\\frac{1}{2}\\left(x - 3\\right)+4[\/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

25.\u00a0[latex]\\frac{{\\left(x+3\\right)}^{2}}{{5}^{2}}-\\frac{{\\left(y - 4\\right)}^{2}}{{2}^{2}}=1[\/latex]; vertices: [latex]\\left(2,4\\right),\\left(-8,4\\right)[\/latex]; foci: [latex]\\left(-3+\\sqrt{29},4\\right),\\left(-3-\\sqrt{29},4\\right)[\/latex]; asymptotes: [latex]y=\\frac{2}{5}\\left(x+3\\right)+4,y=-\\frac{2}{5}\\left(x+3\\right)+4[\/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.
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35.
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37.
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39.
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41.
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43.
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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

57.\u00a0[latex]y\\left(x\\right)=3\\sqrt{{x}^{2}+1},y\\left(x\\right)=-3\\sqrt{{x}^{2}+1}[\/latex]
\n\"\"<\/p>\n

59.\u00a0[latex]y\\left(x\\right)=1+2\\sqrt{{x}^{2}+4x+5},y\\left(x\\right)=1 - 2\\sqrt{{x}^{2}+4x+5}[\/latex]
\n\"\"<\/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\n\t\t\t

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Candela Citations<\/h3>\n\t\t\t\t\t
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CC licensed content, Specific attribution<\/div>