## Problem Set 68: Rotation of Axes

1. What effect does the $xy$ term have on the graph of a conic section?

2. If the equation of a conic section is written in the form $A{x}^{2}+B{y}^{2}+Cx+Dy+E=0$ and $AB=0$, what can we conclude?

3. If the equation of a conic section is written in the form $A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0$, and ${B}^{2}-4AC>0$, what can we conclude?

4. Given the equation $a{x}^{2}+4x+3{y}^{2}-12=0$, what can we conclude if $a>0?$

5. For the equation $A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0$, the value of $\theta$ that satisfies $\cot \left(2\theta \right)=\frac{A-C}{B}$ gives us what information?

For the following exercises, determine which conic section is represented based on the given equation.

6. $9{x}^{2}+4{y}^{2}+72x+36y - 500=0$

7. ${x}^{2}-10x+4y - 10=0$

8. $2{x}^{2}-2{y}^{2}+4x - 6y - 2=0$

9. $4{x}^{2}-{y}^{2}+8x - 1=0$

10. $4{y}^{2}-5x+9y+1=0$

11. $2{x}^{2}+3{y}^{2}-8x - 12y+2=0$

12. $4{x}^{2}+9xy+4{y}^{2}-36y - 125=0$

13. $3{x}^{2}+6xy+3{y}^{2}-36y - 125=0$

14. $-3{x}^{2}+3\sqrt{3}xy - 4{y}^{2}+9=0$

15. $2{x}^{2}+4\sqrt{3}xy+6{y}^{2}-6x - 3=0$

16. $-{x}^{2}+4\sqrt{2}xy+2{y}^{2}-2y+1=0$

17. $8{x}^{2}+4\sqrt{2}xy+4{y}^{2}-10x+1=0$

For the following exercises, find a new representation of the given equation after rotating through the given angle.

18. $3{x}^{2}+xy+3{y}^{2}-5=0,\theta =45^\circ$

19. $4{x}^{2}-xy+4{y}^{2}-2=0,\theta =45^\circ$

20. $2{x}^{2}+8xy - 1=0,\theta =30^\circ$

21. $-2{x}^{2}+8xy+1=0,\theta =45^\circ$

22. $4{x}^{2}+\sqrt{2}xy+4{y}^{2}+y+2=0,\theta =45^\circ$

For the following exercises, determine the angle $\theta$ that will eliminate the $xy$ term and write the corresponding equation without the $xy$ term.

23. ${x}^{2}+3\sqrt{3}xy+4{y}^{2}+y - 2=0$

24. $4{x}^{2}+2\sqrt{3}xy+6{y}^{2}+y - 2=0$

25. $9{x}^{2}-3\sqrt{3}xy+6{y}^{2}+4y - 3=0$

26. $-3{x}^{2}-\sqrt{3}xy - 2{y}^{2}-x=0$

27. $16{x}^{2}+24xy+9{y}^{2}+6x - 6y+2=0$

28. ${x}^{2}+4xy+4{y}^{2}+3x - 2=0$

29. ${x}^{2}+4xy+{y}^{2}-2x+1=0$

30. $4{x}^{2}-2\sqrt{3}xy+6{y}^{2}-1=0$

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. $y=-{x}^{2},\theta =-{45}^{\circ }$

32. $x={y}^{2},\theta ={45}^{\circ }$

33. $\frac{{x}^{2}}{4}+\frac{{y}^{2}}{1}=1,\theta ={45}^{\circ }$

34. $\frac{{y}^{2}}{16}+\frac{{x}^{2}}{9}=1,\theta ={45}^{\circ }$

35. ${y}^{2}-{x}^{2}=1,\theta ={45}^{\circ }$

36. $y=\frac{{x}^{2}}{2},\theta ={30}^{\circ }$

37. $x={\left(y - 1\right)}^{2},\theta ={30}^{\circ }$

38. $\frac{{x}^{2}}{9}+\frac{{y}^{2}}{4}=1,\theta ={30}^{\circ }$

For the following exercises, graph the equation relative to the \begin{align}{x}^{\prime }{y}^{\prime }\end{align} system in which the equation has no \begin{align}{x}^{\prime }{y}^{\prime }\end{align} term.

39. $xy=9$

40. ${x}^{2}+10xy+{y}^{2}-6=0$

41. ${x}^{2}-10xy+{y}^{2}-24=0$

42. $4{x}^{2}-3\sqrt{3}xy+{y}^{2}-22=0$

43. $6{x}^{2}+2\sqrt{3}xy+4{y}^{2}-21=0$

44. $11{x}^{2}+10\sqrt{3}xy+{y}^{2}-64=0$

45. $21{x}^{2}+2\sqrt{3}xy+19{y}^{2}-18=0$

46. $16{x}^{2}+24xy+9{y}^{2}-130x+90y=0$

47. $16{x}^{2}+24xy+9{y}^{2}-60x+80y=0$

48. $13{x}^{2}-6\sqrt{3}xy+7{y}^{2}-16=0$

49. $4{x}^{2}-4xy+{y}^{2}-8\sqrt{5}x - 16\sqrt{5}y=0$

For the following exercises, determine the angle of rotation in order to eliminate the $xy$ term. Then graph the new set of axes.

50. $6{x}^{2}-5\sqrt{3}xy+{y}^{2}+10x - 12y=0$

51. $6{x}^{2}-5xy+6{y}^{2}+20x-y=0$

52. $6{x}^{2}-8\sqrt{3}xy+14{y}^{2}+10x - 3y=0$

53. $4{x}^{2}+6\sqrt{3}xy+10{y}^{2}+20x - 40y=0$

54. $8{x}^{2}+3xy+4{y}^{2}+2x - 4=0$

55. $16{x}^{2}+24xy+9{y}^{2}+20x - 44y=0$

For the following exercises, determine the value of $k$ based on the given equation.

56. Given $4{x}^{2}+kxy+16{y}^{2}+8x+24y - 48=0$, find $k$ for the graph to be a parabola.

57. Given $2{x}^{2}+kxy+12{y}^{2}+10x - 16y+28=0$, find $k$ for the graph to be an ellipse.

58. Given $3{x}^{2}+kxy+4{y}^{2}-6x+20y+128=0$, find $k$ for the graph to be a hyperbola.

59. Given $k{x}^{2}+8xy+8{y}^{2}-12x+16y+18=0$, find $k$ for the graph to be a parabola.

60. Given $6{x}^{2}+12xy+k{y}^{2}+16x+10y+4=0$, find $k$ for the graph to be an ellipse.