Solutions to Odd-Numbered Exercises

1. The $xy$ term causes a rotation of the graph to occur.

3. The conic section is a hyperbola.

5. It gives the angle of rotation of the axes in order to eliminate the $xy$ term.

7. $AB=0$, parabola

9. $AB=-4<0$, hyperbola 11. $AB=6>0$, ellipse

13. ${B}^{2}-4AC=0$, parabola

15. ${B}^{2}-4AC=0$, parabola

17. ${B}^{2}-4AC=-96<0$, ellipse 19. $7{{x}^{\prime }}^{2}+9{{y}^{\prime }}^{2}-4=0$ 21. $3{{x}^{\prime }}^{2}+2{x}^{\prime }{y}^{\prime }-5{{y}^{\prime }}^{2}+1=0$ 23. $\theta ={60}^{\circ },11{{x}^{\prime }}^{2}-{{y}^{\prime }}^{2}+\sqrt{3}{x}^{\prime }+{y}^{\prime }-4=0$ 25. $\theta ={150}^{\circ },21{{x}^{\prime }}^{2}+9{{y}^{\prime }}^{2}+4{x}^{\prime }-4\sqrt{3}{y}^{\prime }-6=0$ 27. $\theta \approx {36.9}^{\circ },125{{x}^{\prime }}^{2}+6{x}^{\prime }-42{y}^{\prime }+10=0$ 29. $\theta ={45}^{\circ },3{{x}^{\prime }}^{2}-{{y}^{\prime }}^{2}-\sqrt{2}{x}^{\prime }+\sqrt{2}{y}^{\prime }+1=0$ 31. $\frac{\sqrt{2}}{2}\left({x}^{\prime }+{y}^{\prime }\right)=\frac{1}{2}{\left({x}^{\prime }-{y}^{\prime }\right)}^{2}$

33. $\frac{{\left({x}^{\prime }-{y}^{\prime }\right)}^{2}}{8}+\frac{{\left({x}^{\prime }+{y}^{\prime }\right)}^{2}}{2}=1$

35. $\frac{{\left({x}^{\prime }+{y}^{\prime }\right)}^{2}}{2}-\frac{{\left({x}^{\prime }-{y}^{\prime }\right)}^{2}}{2}=1$

37. $\frac{\sqrt{3}}{2}{x}^{\prime }-\frac{1}{2}{y}^{\prime }={\left(\frac{1}{2}{x}^{\prime }+\frac{\sqrt{3}}{2}{y}^{\prime }-1\right)}^{2}$

39.

41.

43.

45.

47.

49.

51. $\theta ={45}^{\circ }$

53. $\theta ={60}^{\circ }$

55. $\theta \approx {36.9}^{\circ }$

57. [latex]-4\sqrt{6}