1128 APPENDIX B Rotation of Axes Appendix B Exercises Concept Check Use the summary at the end of the section to predict the type of graph of each second-degree equation. 1. 4x2 + 3y2 + 2xy - 5x = 8 2. x2 + 2xy - 3y2 + 2y = 12 3. 2x2 + 3xy - 4y2 = 0 4. x2 - 2xy + y2 + 4x - 8y = 0 5. 4x2 + 4xy + y2 + 15 = 0 6. x2 - 2xy + y2 - 16 = 0 Concept Check Find the angle of rotation u that will remove the xy-term in each equation. 7. 2x2 + 23xy + y2 + x = 5 8. 423x2 + xy + 323y2 = 10 9. 3x2 + 23xy + 4y2 + 2x - 3y = 12 10. 4x2 + 2xy + 2y2 + x = 7 11. x2 - 4xy + 5y2 = 18 12. 323x2 - 2xy + 23y2 = 25 Find the resulting equation if the axes are rotated through angle u. Graph the equation. See Example 1. 13. x2 - xy + y2 = 6; u = 45° 14. 5y2 + 12xy = 10; u = sin-1¢ 3213 13 ≤ Remove the xy-term from each equation by performing a suitable rotation. Then graph each equation. See Example 2. 15. x2 - 4xy + y2 = -5 16. 3x2 - 2xy + 3y2 = 8 17. 7x2 + 623xy + 13y2 = 64 18. x2 + 2xy + y2 + 422x = 422y 19. 3x2 - 223xy + y2 - 2x = 223y 20. 7x2 + 223xy + 5y2 = 24 Remove the xy-term by rotation. Then translate the axes and sketch the graph. 21. x2 + 3xy + y2 - 522y - 15 = 0 22. x2 - 23xy + 223x - 3y - 3 = 0 23. 4x2 + 4xy + y2 - 24x + 38y - 19 = 0 24. 12x2 + 24xy + 19y2 - 12x - 40y + 31 = 0 25. 16x2 + 24xy + 9y2 - 130x + 90y = 0 26. 9x2 - 6xy + y2 - 12210x - 36210y = 0
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