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Study Guides > College Algebra

Section Exercises

1. What effect does the xyxy 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=0A{x}^{2}+B{y}^{2}+Cx+Dy+E=0 and AB=0AB=0, what can we conclude? 3. If the equation of a conic section is written in the form Ax2+Bxy+Cy2+Dx+Ey+F=0A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0, and B24AC>0{B}^{2}-4AC>0, what can we conclude? 4. Given the equation ax2+4x+3y212=0a{x}^{2}+4x+3{y}^{2}-12=0, what can we conclude if a>0?a>0? 5. For the equation Ax2+Bxy+Cy2+Dx+Ey+F=0A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0, the value of θ\theta that satisfies cot(2θ)=ACB\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. 9x2+4y2+72x+36y500=09{x}^{2}+4{y}^{2}+72x+36y - 500=0 7. x210x+4y10=0{x}^{2}-10x+4y - 10=0 8. 2x22y2+4x6y2=02{x}^{2}-2{y}^{2}+4x - 6y - 2=0 9. 4x2y2+8x1=04{x}^{2}-{y}^{2}+8x - 1=0 10. 4y25x+9y+1=04{y}^{2}-5x+9y+1=0 11. 2x2+3y28x12y+2=02{x}^{2}+3{y}^{2}-8x - 12y+2=0 12. 4x2+9xy+4y236y125=04{x}^{2}+9xy+4{y}^{2}-36y - 125=0 13. 3x2+6xy+3y236y125=03{x}^{2}+6xy+3{y}^{2}-36y - 125=0 14. 3x2+33xy4y2+9=0-3{x}^{2}+3\sqrt{3}xy - 4{y}^{2}+9=0 15. 2x2+43xy+6y26x3=02{x}^{2}+4\sqrt{3}xy+6{y}^{2}-6x - 3=0 16. x2+42xy+2y22y+1=0-{x}^{2}+4\sqrt{2}xy+2{y}^{2}-2y+1=0 17. 8x2+42xy+4y210x+1=08{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. 3x2+xy+3y25=0,θ=453{x}^{2}+xy+3{y}^{2}-5=0,\theta =45^\circ 19. 4x2xy+4y22=0,θ=454{x}^{2}-xy+4{y}^{2}-2=0,\theta =45^\circ 20. 2x2+8xy1=0,θ=302{x}^{2}+8xy - 1=0,\theta =30^\circ 21. 2x2+8xy+1=0,θ=45-2{x}^{2}+8xy+1=0,\theta =45^\circ 22. 4x2+2xy+4y2+y+2=0,θ=454{x}^{2}+\sqrt{2}xy+4{y}^{2}+y+2=0,\theta =45^\circ For the following exercises, determine the angle θ\theta that will eliminate the xyxy term and write the corresponding equation without the xyxy term. 23. x2+33xy+4y2+y2=0{x}^{2}+3\sqrt{3}xy+4{y}^{2}+y - 2=0 24. 4x2+23xy+6y2+y2=04{x}^{2}+2\sqrt{3}xy+6{y}^{2}+y - 2=0 25. 9x233xy+6y2+4y3=09{x}^{2}-3\sqrt{3}xy+6{y}^{2}+4y - 3=0 26. 3x23xy2y2x=0-3{x}^{2}-\sqrt{3}xy - 2{y}^{2}-x=0 27. 16x2+24xy+9y2+6x6y+2=016{x}^{2}+24xy+9{y}^{2}+6x - 6y+2=0 28. x2+4xy+4y2+3x2=0{x}^{2}+4xy+4{y}^{2}+3x - 2=0 29. x2+4xy+y22x+1=0{x}^{2}+4xy+{y}^{2}-2x+1=0 30. 4x223xy+6y21=04{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=x2,θ=45y=-{x}^{2},\theta =-{45}^{\circ } 32. x=y2,θ=45x={y}^{2},\theta ={45}^{\circ } 33. x24+y21=1,θ=45\frac{{x}^{2}}{4}+\frac{{y}^{2}}{1}=1,\theta ={45}^{\circ } 34. y216+x29=1,θ=45\frac{{y}^{2}}{16}+\frac{{x}^{2}}{9}=1,\theta ={45}^{\circ } 35. y2x2=1,θ=45{y}^{2}-{x}^{2}=1,\theta ={45}^{\circ } 36. y=x22,θ=30y=\frac{{x}^{2}}{2},\theta ={30}^{\circ } 37. x=(y1)2,θ=30x={\left(y - 1\right)}^{2},\theta ={30}^{\circ } 38. x29+y24=1,θ=30\frac{{x}^{2}}{9}+\frac{{y}^{2}}{4}=1,\theta ={30}^{\circ } For the following exercises, graph the equation relative to the xy{x}^{\prime }{y}^{\prime } system in which the equation has no xy{x}^{\prime }{y}^{\prime } term. 39. xy=9xy=9 40. x2+10xy+y26=0{x}^{2}+10xy+{y}^{2}-6=0 41. x210xy+y224=0{x}^{2}-10xy+{y}^{2}-24=0 42. 4x233xy+y222=04{x}^{2}-3\sqrt{3}xy+{y}^{2}-22=0 43. 6x2+23xy+4y221=06{x}^{2}+2\sqrt{3}xy+4{y}^{2}-21=0 44. 11x2+103xy+y264=011{x}^{2}+10\sqrt{3}xy+{y}^{2}-64=0 45. 21x2+23xy+19y218=021{x}^{2}+2\sqrt{3}xy+19{y}^{2}-18=0 46. 16x2+24xy+9y2130x+90y=016{x}^{2}+24xy+9{y}^{2}-130x+90y=0 47. 16x2+24xy+9y260x+80y=016{x}^{2}+24xy+9{y}^{2}-60x+80y=0 48. 13x263xy+7y216=013{x}^{2}-6\sqrt{3}xy+7{y}^{2}-16=0 49. 4x24xy+y285x165y=04{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 xyxy term. Then graph the new set of axes. 50. 6x253xy+y2+10x12y=06{x}^{2}-5\sqrt{3}xy+{y}^{2}+10x - 12y=0 51. 6x25xy+6y2+20xy=06{x}^{2}-5xy+6{y}^{2}+20x-y=0 52. 6x283xy+14y2+10x3y=06{x}^{2}-8\sqrt{3}xy+14{y}^{2}+10x - 3y=0 53. 4x2+63xy+10y2+20x40y=04{x}^{2}+6\sqrt{3}xy+10{y}^{2}+20x - 40y=0 54. 8x2+3xy+4y2+2x4=08{x}^{2}+3xy+4{y}^{2}+2x - 4=0 55. 16x2+24xy+9y2+20x44y=016{x}^{2}+24xy+9{y}^{2}+20x - 44y=0 For the following exercises, determine the value of kk based on the given equation. 56. Given 4x2+kxy+16y2+8x+24y48=04{x}^{2}+kxy+16{y}^{2}+8x+24y - 48=0, find kk for the graph to be a parabola. 57. Given 2x2+kxy+12y2+10x16y+28=02{x}^{2}+kxy+12{y}^{2}+10x - 16y+28=0, find kk for the graph to be an ellipse. 58. Given 3x2+kxy+4y26x+20y+128=03{x}^{2}+kxy+4{y}^{2}-6x+20y+128=0, find kk for the graph to be a hyperbola. 59. Given kx2+8xy+8y212x+16y+18=0k{x}^{2}+8xy+8{y}^{2}-12x+16y+18=0, find kk for the graph to be a parabola. 60. Given 6x2+12xy+ky2+16x+10y+4=06{x}^{2}+12xy+k{y}^{2}+16x+10y+4=0, find kk for the graph to be an ellipse.

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