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 Ax2+By2+Cx+Dy+E=0 and AB=0, what can we conclude?
3. If the equation of a conic section is written in the form Ax2+Bxy+Cy2+Dx+Ey+F=0, and B2−4AC>0, what can we conclude?
4. Given the equation ax2+4x+3y2−12=0, what can we conclude if a>0?
5. For the equation Ax2+Bxy+Cy2+Dx+Ey+F=0, the value of θ that satisfies cot(2θ)=BA−C gives us what information?
For the following exercises, determine which conic section is represented based on the given equation.
6. 9x2+4y2+72x+36y−500=0
7. x2−10x+4y−10=0
8. 2x2−2y2+4x−6y−2=0
9. 4x2−y2+8x−1=0
10. 4y2−5x+9y+1=0
11. 2x2+3y2−8x−12y+2=0
12. 4x2+9xy+4y2−36y−125=0
13. 3x2+6xy+3y2−36y−125=0
14. −3x2+33xy−4y2+9=0
15. 2x2+43xy+6y2−6x−3=0
16. −x2+42xy+2y2−2y+1=0
17. 8x2+42xy+4y2−10x+1=0
For the following exercises, find a new representation of the given equation after rotating through the given angle.
18. 3x2+xy+3y2−5=0,θ=45∘
19. 4x2−xy+4y2−2=0,θ=45∘
20. 2x2+8xy−1=0,θ=30∘
21. −2x2+8xy+1=0,θ=45∘
22. 4x2+2xy+4y2+y+2=0,θ=45∘
For the following exercises, determine the angle θ that will eliminate the xy term and write the corresponding equation without the xy term.
23. x2+33xy+4y2+y−2=0
24. 4x2+23xy+6y2+y−2=0
25. 9x2−33xy+6y2+4y−3=0
26. −3x2−3xy−2y2−x=0
27. 16x2+24xy+9y2+6x−6y+2=0
28. x2+4xy+4y2+3x−2=0
29. x2+4xy+y2−2x+1=0
30. 4x2−23xy+6y2−1=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=−x2,θ=−45∘
32. x=y2,θ=45∘
33. 4x2+1y2=1,θ=45∘
34. 16y2+9x2=1,θ=45∘
35. y2−x2=1,θ=45∘
36. y=2x2,θ=30∘
37. x=(y−1)2,θ=30∘
38. 9x2+4y2=1,θ=30∘
For the following exercises, graph the equation relative to the x′y′ system in which the equation has no x′y′ term.
39. xy=9
40. x2+10xy+y2−6=0
41. x2−10xy+y2−24=0
42. 4x2−33xy+y2−22=0
43. 6x2+23xy+4y2−21=0
44. 11x2+103xy+y2−64=0
45. 21x2+23xy+19y2−18=0
46. 16x2+24xy+9y2−130x+90y=0
47. 16x2+24xy+9y2−60x+80y=0
48. 13x2−63xy+7y2−16=0
49. 4x2−4xy+y2−85x−165y=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. 6x2−53xy+y2+10x−12y=0
51. 6x2−5xy+6y2+20x−y=0
52. 6x2−83xy+14y2+10x−3y=0
53. 4x2+63xy+10y2+20x−40y=0
54. 8x2+3xy+4y2+2x−4=0
55. 16x2+24xy+9y2+20x−44y=0
For the following exercises, determine the value of k based on the given equation.
56. Given 4x2+kxy+16y2+8x+24y−48=0, find k for the graph to be a parabola.
57. Given 2x2+kxy+12y2+10x−16y+28=0, find k for the graph to be an ellipse.
58. Given 3x2+kxy+4y2−6x+20y+128=0, find k for the graph to be a hyperbola.
59. Given kx2+8xy+8y2−12x+16y+18=0, find k for the graph to be a parabola.
60. Given 6x2+12xy+ky2+16x+10y+4=0, find k for the graph to be an ellipse.