Solutions
Solutions to Try Its
1. [latex]{x}^{2}+\frac{{y}^{2}}{16}=1[/latex] 2. [latex]\frac{{\left(x - 1\right)}^{2}}{16}+\frac{{\left(y - 3\right)}^{2}}{4}=1[/latex] 3. center: [latex]\left(0,0\right)[/latex]; vertices: [latex]\left(\pm 6,0\right)[/latex]; co-vertices: [latex]\left(0,\pm 2\right)[/latex]; foci: [latex]\left(\pm 4\sqrt{2},0\right)[/latex]
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Solutions to Odd-Numbered Exercises
1. 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. 3. This special case would be a circle. 5. It is symmetric about the x-axis, y-axis, and the origin. 7. yes; [latex]\frac{{x}^{2}}{{3}^{2}}+\frac{{y}^{2}}{{2}^{2}}=1[/latex] 9. yes; [latex]\frac{{x}^{2}}{{\left(\frac{1}{2}\right)}^{2}}+\frac{{y}^{2}}{{\left(\frac{1}{3}\right)}^{2}}=1[/latex] 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]. 13. [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]. 15. [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]. 17. [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]. 19. [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]. 21. [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]. 23. [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]. 25. [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]. 27. Foci [latex]\left(-3,-1+\sqrt{11}\right),\left(-3,-1-\sqrt{11}\right)[/latex] 29. Focus [latex]\left(0,0\right)[/latex] 31. Foci [latex]\left(-10,30\right),\left(-10,-30\right)[/latex] 33. 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]
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