1. What effect does the [latex]xy[\/latex] term have on the graph of a conic section?<\/p>\n
2. If the equation of a conic section is written in the form [latex]A{x}^{2}+B{y}^{2}+Cx+Dy+E=0[\/latex] and [latex]AB=0[\/latex], what can we conclude?<\/p>\n
3. If the equation of a conic section is written in the form [latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex], and [latex]{B}^{2}-4AC>0[\/latex], what can we conclude?<\/p>\n
4. Given the equation [latex]a{x}^{2}+4x+3{y}^{2}-12=0[\/latex], what can we conclude if [latex]a>0?[\/latex]<\/p>\n
5. For the equation [latex]A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0[\/latex], the value of [latex]\\theta[\/latex] that satisfies [latex]\\cot \\left(2\\theta \\right)=\\frac{A-C}{B}[\/latex] gives us what information?<\/p>\n
For the following exercises, determine which conic section is represented based on the given equation.<\/p>\n
6. [latex]9{x}^{2}+4{y}^{2}+72x+36y - 500=0[\/latex]<\/p>\n
7. [latex]{x}^{2}-10x+4y - 10=0[\/latex]<\/p>\n
8. [latex]2{x}^{2}-2{y}^{2}+4x - 6y - 2=0[\/latex]<\/p>\n
9. [latex]4{x}^{2}-{y}^{2}+8x - 1=0[\/latex]<\/p>\n
10. [latex]4{y}^{2}-5x+9y+1=0[\/latex]<\/p>\n
11. [latex]2{x}^{2}+3{y}^{2}-8x - 12y+2=0[\/latex]<\/p>\n
12. [latex]4{x}^{2}+9xy+4{y}^{2}-36y - 125=0[\/latex]<\/p>\n
13. [latex]3{x}^{2}+6xy+3{y}^{2}-36y - 125=0[\/latex]<\/p>\n
14. [latex]-3{x}^{2}+3\\sqrt{3}xy - 4{y}^{2}+9=0[\/latex]<\/p>\n
15. [latex]2{x}^{2}+4\\sqrt{3}xy+6{y}^{2}-6x - 3=0[\/latex]<\/p>\n
16. [latex]-{x}^{2}+4\\sqrt{2}xy+2{y}^{2}-2y+1=0[\/latex]<\/p>\n
17. [latex]8{x}^{2}+4\\sqrt{2}xy+4{y}^{2}-10x+1=0[\/latex]<\/p>\n
For the following exercises, find a new representation of the given equation after rotating through the given angle.<\/p>\n
18. [latex]3{x}^{2}+xy+3{y}^{2}-5=0,\\theta =45^\\circ[\/latex]<\/p>\n
19. [latex]4{x}^{2}-xy+4{y}^{2}-2=0,\\theta =45^\\circ[\/latex]<\/p>\n
20. [latex]2{x}^{2}+8xy - 1=0,\\theta =30^\\circ[\/latex]<\/p>\n
21. [latex]-2{x}^{2}+8xy+1=0,\\theta =45^\\circ[\/latex]<\/p>\n
22. [latex]4{x}^{2}+\\sqrt{2}xy+4{y}^{2}+y+2=0,\\theta =45^\\circ[\/latex]<\/p>\n
For the following exercises, determine the angle [latex]\\theta[\/latex] that will eliminate the [latex]xy[\/latex] term and write the corresponding equation without the [latex]xy[\/latex] term.<\/p>\n
23. [latex]{x}^{2}+3\\sqrt{3}xy+4{y}^{2}+y - 2=0[\/latex]<\/p>\n
24. [latex]4{x}^{2}+2\\sqrt{3}xy+6{y}^{2}+y - 2=0[\/latex]<\/p>\n
25. [latex]9{x}^{2}-3\\sqrt{3}xy+6{y}^{2}+4y - 3=0[\/latex]<\/p>\n
26. [latex]-3{x}^{2}-\\sqrt{3}xy - 2{y}^{2}-x=0[\/latex]<\/p>\n
27. [latex]16{x}^{2}+24xy+9{y}^{2}+6x - 6y+2=0[\/latex]<\/p>\n
28. [latex]{x}^{2}+4xy+4{y}^{2}+3x - 2=0[\/latex]<\/p>\n
29. [latex]{x}^{2}+4xy+{y}^{2}-2x+1=0[\/latex]<\/p>\n
30. [latex]4{x}^{2}-2\\sqrt{3}xy+6{y}^{2}-1=0[\/latex]<\/p>\n
For the following exercises, rotate through the given angle based on the given equation. Give the new equation and graph the original and rotated equation.<\/p>\n
31. [latex]y=-{x}^{2},\\theta =-{45}^{\\circ }[\/latex]<\/p>\n
32. [latex]x={y}^{2},\\theta ={45}^{\\circ }[\/latex]<\/p>\n
33. [latex]\\frac{{x}^{2}}{4}+\\frac{{y}^{2}}{1}=1,\\theta ={45}^{\\circ }[\/latex]<\/p>\n
34. [latex]\\frac{{y}^{2}}{16}+\\frac{{x}^{2}}{9}=1,\\theta ={45}^{\\circ }[\/latex]<\/p>\n
35. [latex]{y}^{2}-{x}^{2}=1,\\theta ={45}^{\\circ }[\/latex]<\/p>\n
36. [latex]y=\\frac{{x}^{2}}{2},\\theta ={30}^{\\circ }[\/latex]<\/p>\n
37. [latex]x={\\left(y - 1\\right)}^{2},\\theta ={30}^{\\circ }[\/latex]<\/p>\n
38. [latex]\\frac{{x}^{2}}{9}+\\frac{{y}^{2}}{4}=1,\\theta ={30}^{\\circ }[\/latex]<\/p>\n
For the following exercises, graph the equation relative to the [latex]\\begin{align}{x}^{\\prime }{y}^{\\prime }\\end{align}[\/latex] system in which the equation has no [latex]\\begin{align}{x}^{\\prime }{y}^{\\prime }\\end{align}[\/latex] term.<\/p>\n
39. [latex]xy=9[\/latex]<\/p>\n
40. [latex]{x}^{2}+10xy+{y}^{2}-6=0[\/latex]<\/p>\n
41. [latex]{x}^{2}-10xy+{y}^{2}-24=0[\/latex]<\/p>\n
42. [latex]4{x}^{2}-3\\sqrt{3}xy+{y}^{2}-22=0[\/latex]<\/p>\n
43. [latex]6{x}^{2}+2\\sqrt{3}xy+4{y}^{2}-21=0[\/latex]<\/p>\n
44. [latex]11{x}^{2}+10\\sqrt{3}xy+{y}^{2}-64=0[\/latex]<\/p>\n
45. [latex]21{x}^{2}+2\\sqrt{3}xy+19{y}^{2}-18=0[\/latex]<\/p>\n
46. [latex]16{x}^{2}+24xy+9{y}^{2}-130x+90y=0[\/latex]<\/p>\n
47. [latex]16{x}^{2}+24xy+9{y}^{2}-60x+80y=0[\/latex]<\/p>\n
48. [latex]13{x}^{2}-6\\sqrt{3}xy+7{y}^{2}-16=0[\/latex]<\/p>\n
49. [latex]4{x}^{2}-4xy+{y}^{2}-8\\sqrt{5}x - 16\\sqrt{5}y=0[\/latex]<\/p>\n
For the following exercises, determine the angle of rotation in order to eliminate the [latex]xy[\/latex] term. Then graph the new set of axes.<\/p>\n
50. [latex]6{x}^{2}-5\\sqrt{3}xy+{y}^{2}+10x - 12y=0[\/latex]<\/p>\n
51. [latex]6{x}^{2}-5xy+6{y}^{2}+20x-y=0[\/latex]<\/p>\n
52. [latex]6{x}^{2}-8\\sqrt{3}xy+14{y}^{2}+10x - 3y=0[\/latex]<\/p>\n
53. [latex]4{x}^{2}+6\\sqrt{3}xy+10{y}^{2}+20x - 40y=0[\/latex]<\/p>\n
54. [latex]8{x}^{2}+3xy+4{y}^{2}+2x - 4=0[\/latex]<\/p>\n
55. [latex]16{x}^{2}+24xy+9{y}^{2}+20x - 44y=0[\/latex]<\/p>\n
For the following exercises, determine the value of [latex]k[\/latex] based on the given equation.<\/p>\n
56. Given [latex]4{x}^{2}+kxy+16{y}^{2}+8x+24y - 48=0[\/latex], find [latex]k[\/latex] for the graph to be a parabola.<\/p>\n
57. Given [latex]2{x}^{2}+kxy+12{y}^{2}+10x - 16y+28=0[\/latex], find [latex]k[\/latex] for the graph to be an ellipse.<\/p>\n
58. Given [latex]3{x}^{2}+kxy+4{y}^{2}-6x+20y+128=0[\/latex], find [latex]k[\/latex] for the graph to be a hyperbola.<\/p>\n
59. Given [latex]k{x}^{2}+8xy+8{y}^{2}-12x+16y+18=0[\/latex], find [latex]k[\/latex] for the graph to be a parabola.<\/p>\n
60. Given [latex]6{x}^{2}+12xy+k{y}^{2}+16x+10y+4=0[\/latex], find [latex]k[\/latex] for the graph to be an ellipse.<\/p>\n\n\t\t\t Candela Citations<\/h3>\n\t\t\t\t\t