1. What effect does the [latex]xy[/latex] term have on the graph of a conic section?
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?
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?
4. Given the equation [latex]a{x}^{2}+4x+3{y}^{2}-12=0[/latex], what can we conclude if [latex]a>0?[/latex]
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?
For the following exercises, determine which conic section is represented based on the given equation.
6. [latex]9{x}^{2}+4{y}^{2}+72x+36y - 500=0[/latex]
7. [latex]{x}^{2}-10x+4y - 10=0[/latex]
8. [latex]2{x}^{2}-2{y}^{2}+4x - 6y - 2=0[/latex]
9. [latex]4{x}^{2}-{y}^{2}+8x - 1=0[/latex]
10. [latex]4{y}^{2}-5x+9y+1=0[/latex]
11. [latex]2{x}^{2}+3{y}^{2}-8x - 12y+2=0[/latex]
12. [latex]4{x}^{2}+9xy+4{y}^{2}-36y - 125=0[/latex]
13. [latex]3{x}^{2}+6xy+3{y}^{2}-36y - 125=0[/latex]
14. [latex]-3{x}^{2}+3\sqrt{3}xy - 4{y}^{2}+9=0[/latex]
15. [latex]2{x}^{2}+4\sqrt{3}xy+6{y}^{2}-6x - 3=0[/latex]
16. [latex]-{x}^{2}+4\sqrt{2}xy+2{y}^{2}-2y+1=0[/latex]
17. [latex]8{x}^{2}+4\sqrt{2}xy+4{y}^{2}-10x+1=0[/latex]
For the following exercises, find a new representation of the given equation after rotating through the given angle.
18. [latex]3{x}^{2}+xy+3{y}^{2}-5=0,\theta =45^\circ [/latex]
19. [latex]4{x}^{2}-xy+4{y}^{2}-2=0,\theta =45^\circ [/latex]
20. [latex]2{x}^{2}+8xy - 1=0,\theta =30^\circ [/latex]
21. [latex]-2{x}^{2}+8xy+1=0,\theta =45^\circ [/latex]
22. [latex]4{x}^{2}+\sqrt{2}xy+4{y}^{2}+y+2=0,\theta =45^\circ [/latex]
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.
23. [latex]{x}^{2}+3\sqrt{3}xy+4{y}^{2}+y - 2=0[/latex]
24. [latex]4{x}^{2}+2\sqrt{3}xy+6{y}^{2}+y - 2=0[/latex]
25. [latex]9{x}^{2}-3\sqrt{3}xy+6{y}^{2}+4y - 3=0[/latex]
26. [latex]-3{x}^{2}-\sqrt{3}xy - 2{y}^{2}-x=0[/latex]
27. [latex]16{x}^{2}+24xy+9{y}^{2}+6x - 6y+2=0[/latex]
28. [latex]{x}^{2}+4xy+4{y}^{2}+3x - 2=0[/latex]
29. [latex]{x}^{2}+4xy+{y}^{2}-2x+1=0[/latex]
30. [latex]4{x}^{2}-2\sqrt{3}xy+6{y}^{2}-1=0[/latex]
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.
31. [latex]y=-{x}^{2},\theta =-{45}^{\circ }[/latex]
32. [latex]x={y}^{2},\theta ={45}^{\circ }[/latex]
33. [latex]\frac{{x}^{2}}{4}+\frac{{y}^{2}}{1}=1,\theta ={45}^{\circ }[/latex]
34. [latex]\frac{{y}^{2}}{16}+\frac{{x}^{2}}{9}=1,\theta ={45}^{\circ }[/latex]
35. [latex]{y}^{2}-{x}^{2}=1,\theta ={45}^{\circ }[/latex]
36. [latex]y=\frac{{x}^{2}}{2},\theta ={30}^{\circ }[/latex]
37. [latex]x={\left(y - 1\right)}^{2},\theta ={30}^{\circ }[/latex]
38. [latex]\frac{{x}^{2}}{9}+\frac{{y}^{2}}{4}=1,\theta ={30}^{\circ }[/latex]
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.
39. [latex]xy=9[/latex]
40. [latex]{x}^{2}+10xy+{y}^{2}-6=0[/latex]
41. [latex]{x}^{2}-10xy+{y}^{2}-24=0[/latex]
42. [latex]4{x}^{2}-3\sqrt{3}xy+{y}^{2}-22=0[/latex]
43. [latex]6{x}^{2}+2\sqrt{3}xy+4{y}^{2}-21=0[/latex]
44. [latex]11{x}^{2}+10\sqrt{3}xy+{y}^{2}-64=0[/latex]
45. [latex]21{x}^{2}+2\sqrt{3}xy+19{y}^{2}-18=0[/latex]
46. [latex]16{x}^{2}+24xy+9{y}^{2}-130x+90y=0[/latex]
47. [latex]16{x}^{2}+24xy+9{y}^{2}-60x+80y=0[/latex]
48. [latex]13{x}^{2}-6\sqrt{3}xy+7{y}^{2}-16=0[/latex]
49. [latex]4{x}^{2}-4xy+{y}^{2}-8\sqrt{5}x - 16\sqrt{5}y=0[/latex]
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.
50. [latex]6{x}^{2}-5\sqrt{3}xy+{y}^{2}+10x - 12y=0[/latex]
51. [latex]6{x}^{2}-5xy+6{y}^{2}+20x-y=0[/latex]
52. [latex]6{x}^{2}-8\sqrt{3}xy+14{y}^{2}+10x - 3y=0[/latex]
53. [latex]4{x}^{2}+6\sqrt{3}xy+10{y}^{2}+20x - 40y=0[/latex]
54. [latex]8{x}^{2}+3xy+4{y}^{2}+2x - 4=0[/latex]
55. [latex]16{x}^{2}+24xy+9{y}^{2}+20x - 44y=0[/latex]
For the following exercises, determine the value of [latex]k[/latex] based on the given equation.
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.
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.
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.
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.
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.