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All Textbook Solutions for College Algebra

For the following exercise, given information about the graph of the hyperbola, find its equation. 49. Center: (4,2) ; vertex: (9,2) ; one focus: (4+26,2) .For the following exercise, given information about the graph of the hyperbola, find its equation. 50. Center: (3,5) ; vertex: (3,11) ; one focus: (3,5+210) .For the following exercise, given information about the graph of the hyperbola, find its equation. 51 For the following exercise, given information about the graph of the hyperbola, find its equation. 52.For the following exercise, given information about the graph of the hyperbola, find its equation. 53.For the following exercise, given information about the graph of the hyperbola, find its equation. 54.For the following exercise, given information about the graph of the hyperbola, find its equation. 55.For the following exercises, express the equation for the hyperbola as two functions, withy as a function of x. Express as simply as possible. Use a graphing calculator to sketch the graph of the two functions on the same axes. 56. x24y29=1 For the following exercises, express the equation for the hyperbola as two functions, withy as a function of x. Express as simply as possible. Use a graphing calculator to sketch the graph of the two functions on the same axes. 57. y29x21=1 For the following exercises, express the equation for the hyperbola as two functions, withy as a function of x. Express as simply as possible. Use a graphing calculator to sketch the graph of the two functions on the same axes. 58. (x2)216(y+3)225=1 For the following exercises, express the equation for the hyperbola as two functions, withy as a function of x. Express as simply as possible. Use a graphing calculator to sketch the graph of the two functions on the same axes. 59. 4x216x+y22y19=0 For the following exercises, express the equation for the hyperbola as two functions, withy as a function of x. Express as simply as possible. Use a graphing calculator to sketch the graph of the two functions on the same axes. 60. 4x224xy24y+16=0 For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph. 61. The hedge will follow the asymptotes y=xandy=x , and its closest distance to the center fountain is 5 yards.For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph. 62. The hedge will follow the asymptotes y=2xandy=2x , and its closest distance to the center fountain is 6 yards.For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph. 63. The hedge will follow the asymptotes y=12x and y=12x , and its closest distance to the center fountain is 10 yards.For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph. 64. The hedge will follow the asymptotes y=23x and y=23x , and its closest distance to the center fountain is 12 yards.For the following exercises, a hedge is to be constructed in the shape of a hyperbola near a fountain at the center of the yard. Find the equation of the hyperbola and sketch the graph. 65. The hedge will follow the asymptotes y=34x and y=34x and its closest distance to the center fountain is 20 yards.For the following exercises, assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the objects path. Give the equation of the flight path of each object using the given information. 66. The object enters along a path approximated by the line y=x2 and passes within 1 au (astronomical unit) of the sun at its closest approach, so that the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line y=-x+2 .For the following exercises, assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the objects path. Give the equation of the flight path of each object using the given information. 67. The object enters along a path approximated by the line y=2x2 and passes within 0.5 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line y=2x+2 .For the following exercises, assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the objects path. Give the equation of the flight path of each object using the given information. 68. The object enters along a path approximated by the line y=0.5x+2 and passes within 1 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line y=0.5x2 .For the following exercises, assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the objects path. Give the equation of the flight path of each object using the given information. 69. The object enters along a path approximated by the line y=13x1 and passes within 1 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line y=13x+1 .For the following exercises, assume an object enters our solar system and we want to graph its path on a coordinate system with the sun at the origin and the x-axis as the axis of symmetry for the objects path. Give the equation of the flight path of each object using the given information. 70. The object enters along a path approximated by the line y=3x9 and passes within 1 au of the sun at its closest approach, so the sun is one focus of the hyperbola. It then departs the solar system along a path approximated by the line y=3x+9 .Graph y2=16x . Identify and label the focus, directrix, and endpoints of the latus rectum.Graph x2=8y . Identify and label the focus, directrix, and endpoints of the latus rectum.What is the equation for the parabola with focus (0,72) and directrix y=72 ?Graph (y+1)2=4(x8) . Identify and label the vertex, axis of symmetry focus, directrix. and endpoints of the latus rectum.Graph (x+2)2=20(y3) . Identify and label the vertex, axis of symmetry, focus. directrix, and endpoints of the latus rectum.Balcony-sized solar cookers have been designed for families living in India. The top of a dish has a diameter of 1,600 mm. The sun’s rays reflect off the parabolic mirror toward the “cooker”. which is placed 320 mm from the base. a. Find an equation that models a cross-section of the solar cooker. Assume that the vertex of the parabolic mirror is the origin of the coordinate plane. and that the parabola opens to the right (i.e., has the x-axis as its axis of symmetry). b. Use the equation found in part (a) to find the depth of the cooker.Define a parabola in terms of its focus and directrix.If the equation of a parabola is written in standard form and p is positive and the directrix is a vertical line, then what can we conclude about its graph?If the equation of a parabola is written in standard form and p is negative and the directrix is a horizontal line, then what can we conclude about its graph?What is the effect on the graph of a parabola if its equation in standard form has increasing values of p?As the graph of a parabola becomes wider, what will happen to the distance between the focus and directrix?For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form. 6. y2=4x2 For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form. 7. y=4x2 For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form. 8. 3x26y2=12 For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form. 9. (y3)2=8(x2)For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form. 10. y2+12x6y51=0 For the following exercises, determine whether the given equation is a parabola. If so, rewrite the equation in standard form. 11. x=8y2 For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V),focus (F),and directrix (d) of the parabola. 12. y=14x2 For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V),focus (F),and directrix (d) of the parabola. 13. y=4x2 For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V),focus (F),and directrix (d) of the parabola. 14. x=18y2 For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V),focus (F),and directrix (d) of the parabola. 15. x=36y2 For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V),focus (F),and directrix (d) of the parabola. 16. x=136y2 For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V),focus (F),and directrix (d) of the parabola. 17. (x1)2=4(y1)For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola. 18. (y2)2=45(x+4)For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola. 19. (y4)2=2(x+3)For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola. 20. (x+1)2=2(y+4)For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola. 21. (x+4)2=24(y+1)For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola. 22. (y+4)2=16(x+4)For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola. 23. y2+12x6y+21=0 For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola. 24. x24x24y+28=0 For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola. 25. 5x250x4y+113=0 For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola. 26. y224x+4y68=0 For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola. 27. x24x+2y6=0 For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola. 28. y26y+12x3=0 For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola. 29. 3y24x6y+23=0 For the following exercises, rewrite the given equation in standard form, and then determine the vertex (V), focus (F), and directrix (d) of the parabola. 30. x2+4x+8y4=0 For the following exercises, graph the parabola, labeling the focus and the directrix. 31. x=18y2 For the following exercises, graph the parabola, labeling the focus and the directrix. 32. y=36x2 For the following exercises, graph the parabola, labeling the focus and the directrix. 33. y=136x2 For the following exercises, graph the parabola, labeling the focus and the directrix. 34. y=9x2 For the following exercises, graph the parabola, labeling the focus and the directrix. 36. (y2)2=43(x+2)For the following exercises, graph the parabola, labeling the focus and the directrix. 36. 5(x+5)2=4(y+5)For the following exercises, graph the parabola, labeling the focus and the directrix. 37. 6(y+5)2=4(x4)For the following exercises, graph the parabola, labeling the focus and the directrix. 38. y26y8x+1=0 For the following exercises, graph the parabola, labeling the focus and the directrix. 39. x2+8x+4y+20=0 For the following exercises, graph the parabola, labeling the focus and the directrix. 40. 3x2+30x4y+95=0 For the following exercises, graph the parabola, labeling the focus and the directrix. 41. y28x+10y+9=0 For the following exercises, graph the parabola, labeling the focus and the directrix. 42. x2+4x+2y+2=0 For the following exercises, graph the parabola, labeling the focus and the directrix. 43. y2+2y12x+61=0 For the following exercises, graph the parabola, labeling the focus and the directrix. 44. 2x2+8x4y24=0 For the following exercises, find the equation of the parabola given information about its graph. 45. Vertex is (0,0) ; directrix is y=4 , focus is (0,4) .For the following exercises, find the equation of the parabola given information about its graph. 46. Vertex is (0,0) ; directrix is x=4 , focus is (4,0) .For the following exercises, find the equation of the parabola given information about its graph. 47. Vertex is (2,2) ; directrix is x=22, focus is (2+2,2)For the following exercises, find the equation of the parabola given information about its graph. 48. Vertex is (2,3) ; directrix is x=72 focus is (12,3) .For the following exercises, find the equation of the parabola given information about its graph. 49. Vertex is (2,3) ; directrix is x=22 , focus is (0,3)For the following exercises, find the equation of the parabola given information about its graph. 50. Vertex is (1,2) ; directrix is y=113 , focus is (1,13) .For the following exercises, find the equation of the parabola given information about its graph. 51.For the following exercises, find the equation of the parabola given information about its graph. 52.For the following exercises, find the equation of the parabola given information about its graph.For the following exercises, find the equation of the parabola given information about its graph.For the following exercises, find the equation of the parabola given information about its graph.For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation. 56. V(0,0) , Endpoints (2,1),(2,1)For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation. 57. V(0,0) , Endpoints (2,4),(2,4)For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation. 58. V(1,2) , Endpoints (5,5),(7,5)For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation. 59. V(3,1) , Endpoints (0,5),(0,7)For the following exercises, the vertex and endpoints of the latus rectum of a parabola are given. Find the equation. 60. V(4,3) , Endpoints (5,72),(3,72)The mirror in an automobile headlight has a parabolic cross-section with the light bulb at the focus. On a schematic, the equation of the parabola is given as x2=4y . At what coordinates should you place the light bulb?If we want to construct the mirror from the previous exercise such that the focus is located at (0,0.25) , what should the equation of the parabola be?A satellite dish is shaped like a paraboloid of revolution. This means that it can be formed by rotating a parabola around its axis of symmetry. The receiver is to be located at the focus. If the dish is 12 feet across at its opening and 4 feet deep at its center, where should the receiver be placed?Consider the satellite dish from the previous exercise. If the dish is 8 feet across at the opening and 2 feet deep, where should we place the receiver?A searchlight is shaped like a paraboloid of revolution. A light source is located 1 foot from the base along the axis of symmetry, lithe opening of the searchlight is 3 feet across, find the depth.If the searchlight from the previous exercise has the light source located 6 inches from the base along the axis of symmetry and the opening is 4 feet, find the depth.An arch is in the shape of a parabola. It has a span of 100 feet and a maximum height of 20 feet. Find the equation of the parabola, and determine the height of the arch 40 feet from the center.If the arch from the previous exercise has a span of 160 feet and a maximum height of 40 feet, find the equation of the parabola, and determine the distance from the center at which the height is 20 feet.An object is projected so as to follow a parabolic path given by y=x2+96x , where x is the horizontal distance traveled in feet and y is the height. Determine the maximum height the object reaches.For the object from the previous exercise, assume the path followed is given by y=0.5x2+80x . Determine how far along the horizontal the object traveled to reach maximum height.Identify the graph of each of the following nondegenerate conic sections. a. 16y2x2+x4y9=0 b. 16x2+4y2+16x+49y81=0 Rewrite the 13x263xy+7y2=16 in the x’y’system without the x’y’ term.Identify the conic for each of the following without rotating axes. a. x29xy+3y212=0 b. 10x29xy+4y24=0 What effect does the xyterm have on the graph of a conic section?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?If the equation of a conic section is written in the form Ax2+Bxy+Cy2+Dx+Ey+F=0 , and B24AC0 , what can we conclude?Given the equation ax2+4x+3y212=0 , what can we conclude if a0 ?For the equation Ax2+Bxy+Cy2+Dx+Ey+F=0 , the value of that satisfies cot(2)=ACB gives us what information?For the following exercises, determine which conic section is represented based on the given equation. 6. 9x2+4y2+72x+36y500=0 For the following exercises, determine which conic section is represented based on the given equation. 7. x210x+4y10=0 For the following exercises, determine which conic section is represented based on the given equation. 8. 2x22y2+4x6y2=0 For the following exercises, determine which conic section is represented based on the given equation. 9. 4x2y2+8x1=0 For the following exercises, determine which conic section is represented based on the given equation. 10. 4y25x+9y+1=0 For the following exercises, determine which conic section is represented based on the given equation. 11. 2x2+3y28x12y+2=0 For the following exercises, determine which conic section is represented based on the given equation. 12. 4x2+9xy+4y236y125=0 For the following exercises, determine which conic section is represented based on the given equation. 13. 3x2+6xy+3y236y125=0 For the following exercises, determine which conic section is represented based on the given equation. 14. 3x2+33xy4y2+9=0 or the following exercises, determine which conic section is represented based on the given equation. 15. 2x2+43xy+6y26x3=0 For the following exercises, determine which conic section is represented based on the given equation. 16. x2+42xy+2y22y+1=0 For the following exercises, determine which conic section is represented based on the given equation. 17. 8x2+42xy+4y210x+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,=45 For the following exercises, find a new representation of the given equation after rotating through the given angle. 19. 4x2xy+4y22=0,=45 For the following exercises, find a new representation of the given equation after rotating through the given angle. 20. 2x2+8xy1=0,=30 For the following exercises, find a new representation of the given equation after rotating through the given angle. 21. 2x2+8xy+1=0,=45 For the following exercises, find a new representation of the given equation after rotating through the given angle. 22. 4x2+2xy+4y2+y+2=0,=45 For the following exercises, determine the angle that will eliminate the xyterm and write the corresponding equation without the xyterm. 23. x2+33xy+4y2+y2=0 For the following exercises, determine the angle that will eliminate the xyterm and write the corresponding equation without the xyterm. 24. 4x2+23xy+6y2+y2=0 For the following exercises, determine the angle that will eliminate the xyterm and write the corresponding equation without the xyterm. 25. 9x233xy+6y2+4y3=0 For the following exercises, determine the angle that will eliminate the xyterm and write the corresponding equation without the xyterm. 26. 3x23xy2y2x=0 For the following exercises, determine the angle that will eliminate the xyterm and write the corresponding equation without the xyterm. 27. 16x2+24xy+9y2+6x6y+2=0 For the following exercises, determine the angle that will eliminate the xyterm and write the corresponding equation without the xyterm. 28. x2+4xy+4y2+3x2=0 For the following exercises determine the angle that will eliminate the xyterm and write the corresponding equation without the xyterm. 29. x2+4xy+y22x+1=0 For the following exercises determine the angle that will eliminate the xyterm and write the corresponding equation without the xyterm. 30. 4x223xy+6y21=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,=45 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. 32. x=y2,=45 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. 33. x24+y21=1,=45 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. 34. y216+x29=1,=45 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. 35. y2x2=1,=45 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. 36. y=x22,=30 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. 37. x=(y1)2,=30 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. 38. x29+y24=1,=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 For the following exercises, graph the equation relative to the x’y’ system in which the equation has no x’y’ term. 40. x2+10xy+y26=0 For the following exercises, graph the equation relative to the x’y’ system in which the equation has no x’y’ term. 41. x210xy+y224=0 For the following exercises, graph the equation relative to the x’y’ system in which the equation has no x’y’ term. 42. 4x233xy+y222=0 For the following exercises, graph the equation relative to the x’y’ system in which the equation has no x’y’ term. 43. 6x2+23xy+4y221=0 For the following exercises, graph the equation relative to the x’y’ system in which the equation has no x’y’ term. 44. 11x2+103xy+y264=0 For the following exercises, graph the equation relative to the x’y’ system in which the equation has no x’y’ term. 45. 21x2+23xy+19y218=0 For the following exercises, graph the equation relative to the x’y’ system in which the equation has no x’y’ term. 46. 16x2+24xy+9y2130x+90y=0 For the following exercises, graph the equation relative to the x’y’ system in which the equation has no x’y’ term. 47. 16x2+24xy+9y260x+80y=0 For the following exercises, graph the equation relative to the x’y’ system in which the equation has no x’y’ term. 48. 13x263xy+7y216=0 For the following exercises, graph the equation relative to the x’y’ system in which the equation has no x’y’ term. 49. 4x24xy+y285x165y=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. 6x253xy+y2+10x12y=0 For the following exercises, determine the angle of rotation in order to eliminate the xy term. Then graph the new set of axes. 51. 6x25xy+6y2+20xy=0 For the following exercises, determine the angle of rotation in order to eliminate the xy term. Then graph the new set of axes. 52. 6x283xy+14y2+10x3y=0 For the following exercises, determine the angle of rotation in order to eliminate the xy term. Then graph the new set of axes. 53. 4x2+63xy+10y2+20x40y=0 For the following exercises, determine the angle of rotation in order to eliminate the xy term. Then graph the new set of axes. 54. 8x2+3xy+4y2+2x4=0 For the following exercises, determine the angle of rotation in order to eliminate the xy term. Then graph the new set of axes. 55. 16x2+24xy+9y2+20x44y=0 For the following exercises, determine the value of k based on the given equation. 56. Given 4x2+kxy+16y2+8x+24y48=0 , find k for the graph to be a parabola.For the following exercises, determine the value of k based on the given equation. 57. Given 2x2+kxy+12y2+10x16y+28=0 , find k for the graph to be an ellipse.For the following exercises, determine the value of k based on the given equation. 58. Given 3x2+kxy+4y26x+20y+128=0 ,find k for the graph to be a hyperbola.For the following exercises, determine the value of k based on the given equation. 59. Given kx2+8xy+8y212x+16y+18=0 , find k for the graph to be a parabola.For the following exercises, determine the value of k based on the given equation. 60. Given 6x2+12xy+ky2+16x+10y+4=0 , find k for the graph to be an ellipse.Identify the conic with focus at the origin, the directrix, and the eccentricity for r=23cos.Graph r=24cos Find the polar form of the conic given a focus at the origin, e=1, and directrix x=1 .Convert the conic r=21+2cos to rectangular form.Explain how eccentricity determines which conic section is given.If a conic section is written as a polar equation, what must be true of the denominator?If a conic section is written as a polar equation, and the denominator involves sin , what conclusion can be drawn about the directrix?If the directrix of a conic section is perpendicular to the polar axis, what do we know about the equation of the graph?What do we know about the focus/foci of a conic section if it is written as a polar equation?For the following exercises, identity the conic with a focus at the origin, and then give the directrix and eccentricity. 6. r=612cos For the following exercises, identity the conic with a focus at the origin, and then give the directrix and eccentricity. 7. r=344sin For the following exercises, identity the conic with a focus at the origin, and then give the directrix and eccentricity. 8. r=843cos For the following exercises, identity the conic with a focus at the origin, and then give the directrix and eccentricity. 9. r=51+2sin For the following exercises, identity the conic with a focus at the origin, and then give the directrix and eccentricity. 10. r=164+3cos For the following exercises, identity the conic with a focus at the origin, and then give the directrix and eccentricity. 11. r=310+10cos For the following exercises, identity the conic with a focus at the origin, and then give the directrix and eccentricity. 12. r=21cos For the following exercises, identity the conic with a focus at the origin, and then give the directrix and eccentricity. 13. r=47+2cos For the following exercises, identity the conic with a focus at the origin, and then give the directrix and eccentricity. 14. r(1cos)=3 For the following exercises, identity the conic with a focus at the origin, and then give the directrix and eccentricity. 15. r(3+5sin)=11 For the following exercises, identity the conic with a focus at the origin, and then give the directrix and eccentricity. 16. r(45sin)=1 For the following exercises, identity the conic with a focus at the origin, and then give the directrix and eccentricity. 17. r(7+8cos)=7 For the following exercises, convert the polar equation of a conic section to a rectangular equation. 18. r=41+3sin For the following exercises, convert the polar equation of a conic section to a rectangular equation. 19. r=253sin For the following exercises, convert the polar equation of a conic section to a rectangular equation. 20. r=832cos For the following exercises, convert the polar equation of a conic section to a rectangular equation. 21. r=32+5cos For the following exercises, convert the polar equation of a conic section to a rectangular equation. 22. r=42+2sin For the following exercises, convert the polar equation of a conic section to a rectangular equation. 23. r=388cos For the following exercises, convert the polar equation of a conic section to a rectangular equation. 24. r=26+7cos For the following exercises, convert the polar equation of a conic section to a rectangular equation. 25. r=5511sin For the following exercises, convert the polar equation of a conic section to a rectangular equation. 26. r(5+2cos)=6 For the following exercises, convert the polar equation of a conic section to a rectangular equation. 27. r(2cos)=1 For the following exercises, convert the polar equation of a conic section to a rectangular equation. 28. r(2.52.5sin)=5 For the following exercises, convert the polar equation of a conic section to a rectangular equation. 29. r=6sec2+3sec For the following exercises, convert the polar equation of a conic section to a rectangular equation. 30. r=6csc3+2csc For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci. 31. r=52+cos For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci. 32. r=23+3sin For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci. 33. r=1054sin For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci. 34. r=31+2cos For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci. 35. r=845cos For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci. 36. r=344cos For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci. 37. r=21sin For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci. 38. r=63+2sin For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci. 39. r(1+cos)=5 For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci. 40. r(34sin)=9 For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci. 41. r(32sin)=6 For the following exercises, graph the given conic section. If it is a parabola, label the vertex, focus and directrix. If it is an ellipse, label the vertices and foci. If it is a hyperbola, label the vertices and foci. 42. r(64cos)=5 For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. 43. Directrix: x=4;e=15 For the following exercises, find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. 44. Directrix: x=4;e=5 For the following exercises. find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. 45. Directrix: y=2;e=2 For the following exercises. find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. 46. Directrix: y=2;e=12 For the following exercises. find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. 47. Directrix: x=1;e=1 For the following exercises. find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. 48. Directrix: x=1;e=1 For the following exercises. find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. 49. Directrix: x=14;e=72 For the following exercises. find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. 50. Directrix: y=25;e=72 For the following exercises. find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. 51. Directrix: y=4;e=32 For the following exercises. find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. 52. Directrix: x=2;e=83 For the following exercises. find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. 48. Directrix: x=5;e=34 For the following exercises. find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. 48. Directrix: y=2;e=2.5 For the following exercises. find the polar equation of the conic with focus at the origin and the given eccentricity and directrix. 55.Directrix: x=3;e=13 Recall from Rotation of Axes that equations of conics with an xy term have rotated graphs. For the following exercises, express each equation in polar form with r as a function . 56. xy=2 Recall from Rotation of Axes that equations of conics with an xy term have rotated graphs. For the following exercises, express each equation in polar form with r as a function . 57. x2+xy+y2=4 Recall from Rotation of Axes that equations of conics with an xyterm have rotated graphs. For the following exercises, express each equation in polar form with r as a function . 58. 2x2+4xy+2y2=9 Recall from Rotation of Axes that equations of conics with an xyterm have rotated graphs. For the following exercises, express each equation in polar form with r as a function . 59. 16x2+24xy+9y2=4 Recall from Rotation of Axes that equations of conics with an xy term have rotated graphs. For the following exercises, express each equation in polar form with r as a function . 60. 2xy+y=1 For the following exercises. write the equation of the ellipse in standard form. Then identify the center, vertices, and foci. x225+y264=1 For the following exercises, write the equation of the ellipse in standard form. Then identify the center, vertices, and foci. 2. (x2)2100+(y+3)236=1 For the following exercises, write the equation of the ellipse in standard form. Then identify the center, vertices, and foci. 3. 9x2+y2+54x4y+76=0 For the following exercises, write the equation of the ellipse in standard form. Then identify the center, vertices, and foci. 4. 9x2+36y236x+72y+36=0 For the following exercises, graph the ellipse, noting center, vertices, and foci. 5. x236+y29=1 For the following exercises, graph the ellipse, noting center, vertices, and foci. 6. (x4)225+(y+3)249=1 For the following exercises, graph the ellipse, noting center, vertices, and foci. 7. 4x2+y2+16x+4y44=0 For the following exercises, graph the ellipse, noting center, vertices, and foci. 8. 2x2+3y220x+12y+38=0 For the following exercises, use the given information to find the equation for the ellipse. 9. Center at (0,0) , focus at (3,0) , vertex at (5,0)For the following exercises, use the given information to find the equation for the ellipse. 10. Center at (2,2) , vertex at (7,2) , focus at (4,2)For the following exercises, use the given information to find the equation for the ellipse. 11. A whispering gallery is to be constructed such that the foci are located 35 feet from the center. If the length of the gallery is to be 100 feet, what should the height of the ceiling be?For the following exercises. write the equation of the hyperbola in standard form. then give the center, vertices, and foci. 12. x281y29=1 For the following exercises, write the equation of the hyperbola in standard form. Then give the center, vertices, and foci. 13. (y+1)216(x4)236=1 For the following exercises. write the equation of the hyperbola in standard form. Then give the center, vertices, and foci. 14. 9y24x2+54y16x+29=0 For the following exercises, write the equation of the hyperbola in standard form. Then give the center, vertices, and foci. 15. 3x2y212x6y9=0 For the following exercises, graph the hyperbola, labeling vertices and foci. 16. x29y216=1 For the following exercises, graph the hyperbola, labeling vertices and foci. 17. (y1)249(x+1)24=1 For the following exercises, graph the hyperbola, labeling vertices and foci. 18. x24y2+6x+32y91=0 For the following exercises, graph the hyperbola, labeling vertices and foci. 19. 2y2x212y6=0 For the following exercises, find the equation of the hyperbola. 20. Center at (0,0) , vertex at (0,4) , focus at (0,6)For the following exercises, find the equation of the hyperbola. 21. Foci at (3,7)and(7,7) , vertex at (6,7)For the following exercises, write the equation of the parabola in standard form. Then give the vertex, focus, and directrix. 22. y2=12x For the following exercises, write the equation of the parabola in standard form. Then give the vertex, focus, and directrix. 23. (x+2)2=12(y1)For the following exercises, write the equation of the parabola in standard form. Then give the vertex, focus, and directrix. 24. y26y6x3=0 For the following exercises, write the equation of the parabola in standard form, then give the vertex, focus, and directrix. 25. x2+10xy+23=0 For the following exercises, graph the parabola. labeling vertex, focus, and directrix. 26. x2+4y=0 For the following exercises, graph the parabola. labeling vertex, focus, and directrix. 27. (y1)2=12(x+3)For the following exercises, graph the parabola. labeling vertex, focus, and directrix. 28. x28x10y+46=0 For the following exercises, graph the parabola. labeling vertex, focus, and directrix. 29.2y2+12y+6x+15=0 For the following exercises, write the equation of the parabola using the given information. 30. Focus at (4,0) ; directrix is x=4 For the following exercises, write the equation of the parabola using the given information. 31. Focus at (2,98) ; directrix is y=78 For the following exercises, write the equation of the parabola using the given information. 32. A cable TV receiving dish is the shape of a paraboloid of revolution. Find the location of the receiver, which is placed at the locus, if the dish is 5 feet across at its opening and 1.5 feet deep.For the following exercises, determine which of the conic sections is represented. 33. 16x2+24xy+9y2+24x60y+60=0 For the following exercises, determine which of the conic sections is represented. 34. 4x2+14xy+5y2+18x6y+30=0 For the following exercises, determine which of the conic sections is represented. 35. 4x2+xy+2y2+8x26y+9=0 For the following exercises, determine the angle that will eliminate the xyterm, and write the corresponding equation without the xyterm. 36. x2+4xy2y26=0 For the following exercises,determine the angle that will eliminate the xyterm, and write the corresponding equation without the xyterm. 37. x2xy+y26=0 For the following exercises, graph the equation relative to the x’y’ system in which the equation has no x’y’term. 38. 9x224xy+16y280x60y+100=0 For the following exercises, graph the equation relative to the x’y’ system in which the equation has no x’y’ term. 39. x2xy+y22=0 For the following exercises, graph the equation relative to the x’y’ system in which the equation has no x’y’ term. 40. 6x2+24xyy212x+26y+11=0 For the following exercises, given the polar equation of the conic with focus at the origin, identify the eccentricity and directrix. 41. r=1015cos For the following exercises, given the polar equation of the conic with focus at the origin, identify the eccentricity and directrix. 42. r=63+2cos For the following exercises, given the polar equation of the conic with focus at the origin, identify the eccentricity and directrix. 43. r=14+3sin For the following exercises, given the polar equation of the conic with focus at the origin, identify the eccentricity and directrix. 44. r=355sin For the following exercises, graph the conic given in polar form. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse or a hyperbola, label the vertices and foci. 45. r=31sin For the following exercises, graph the conic given in polar form. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse or a hyperbola, label the vertices and foci. 46. r=84+3sin For the following exercises, graph the conic given in polar form. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse or a hyperbola, label the vertices and foci. 47. r=104+5cos For the following exercises, graph the conic given in polar form. If it is a parabola, label the vertex, focus, and directrix. If it is an ellipse or a hyperbola, label the vertices and foci. 48. r=936cos For the following exercises, given information about the graph of a conic with locus at the origin, find the equation in polar form. 49. Directrix is x=3 and eccentricity e=1 For the following exercises, given information about the graph of a conic with locus at the origin, find the equation in polar form. 50. Directrix is y=2 and eccentricity e=4 For the following exercises, write the equation in standard form and state the center, vertices, and foci. x29+y24=1 For the following exercises, write the equation in standard form and state the center, vertices, and foci. 2. 9y2+16x236y+32x92=0 For the following exercises, sketch the graph, identifying the center, vertices, and foci. ( x3)264+( y2)236=1 For the following exercises, sketch the graph. identifying the center, vertices, and foci. 4. 2x2+y2+8x6y7=0 For the following exercises, sketch the graph. identifying the center, vertices, and foci. 5. Write the standard form equation of an ellipse with a center at (1,2) , vertex at (7,2) , and focus at (4,2) .For the following exercises, sketch the graph, identifying the center, vertices, and foci. 6. A whispering gallery is to be constructed with a length of 150 feet. If the foci are to be located 20 feet away from the wall, how high should the ceiling be?For the following exercises, write the equation of the hyperbola in standard form, and give the center, vertices, foci, and asymptotes. x249y281=1 For the following exercises, write the equation of the hyperbola in standard form, and give the center, vertices, foci, and asymptotes. 16y29x2+128y+112=0 For the following exercises, graph the hyperbola, noting its center, vertices, and foci. State the equations of the asymptotes. ( x3)225( y+3)21=1 For the following exercises, graph the hyperbola. noting its center. vertices, and foci. State the equations of the asymptotes. y2x2+4y4x18=0 For the following exercises, graph the hyperbola, noting its center, vertices, and foci. State the equations of the asymptotes. 11. Write the standard form equation of a hyperbola with foci at (1,0)and(1,6) , and a vertex at (1,2) .For the following exercises. write the equation of the parabola in standard form, and give the vertex, focus, and equation of the directrix. y2+10x=0 For the following exercises, write the equation of the parabola in standard form, and give the vertex, focus, and equation of the directrix. 3x212xy+11=0 For the following exercises, graph the parabola, labeling the vertex, focus, and directrix. 14. (x1)2=4(y+3)For the following exercises, graph the parabola, labeling the vertex, focus, and directrix. 15. y2+8x8y+40=0 For the following exercises, graph the parabola, labeling the vertex, focus, and directrix. 16. Write the equation of a parabola with a focus at (2,3) and directrix y=1 .For the following exercises, graph the parabola, labeling the vertex, focus, and directrix. 17. A searchlight is shaped like a paraboloid of revolution. If the light source is located 1.5 feet from the base along the axis of symmetry, and the depth of the searchlight is 3 feet, what should the width of the opening be?For the following exercises, determine which conic section is represented by the given equation, and then determine the angle that will eliminate the xy term. 3x22xy+3y2=4 For the following exercises, determine which conic section is represented by the given equation, and then determine the angle 9 that will eliminate the xy term. x2+4xy+4y2+6x8y=0 For the following exercises, rewrite in the x’y’ system without the x’y’term, and graph the rotated graph. 20. 11x2+103xy+y2=4 For the following exercises, rewrite in the x’y’ system without the x’y’ term, and graph the rotated graph. 21. 16x2+24xy+9y2125x=0 For the following exercises, identify the conic with focus at the origin, and then give the directrix and eccentricity. 22. r=32sin For the following exercises. identify the conic with focus at the origin, and then give the directrix and eccentricity. r=54+6cos For the following exercises, graph the given conic section. hit is a parabola. label vertex, focus, and directrix. If it is an ellipse or a hyperbola, label vertices and foci. 24. r=1248sin For the following exercises, graph the given conic section. hit is a parabola. label vertex, focus, and directrix. If it is an ellipse or a hyperbola, label vertices and foci. 25. r=24+4sin

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