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All Textbook Solutions for Precalculus: Mathematics for Calculus (Standalone Book)

94RE 95RE 96RE 97RE 98RE 99RE 100RE 101RE 102RE 103RE 104RE 105RE 106RE 107RE 108RE 109RE 110RE 111RE 112RE 1T 2T 3T 4T 5T 6T 7T Anne, Barry, and Cathy enter a coffee shop. Anne orders two coffees, one juice, and two doughnuts and pays 6.25. Barry orders one coffee and three doughnuts and pays 3.75. Cathy orders three coffees, one juice, and four doughnuts and pays 9.25. Find the price of coffee, juice, and doughnuts at this coffee shop.9T 10T Only one of the following matrices has an inverse. Find the determinant of each matrix, and use the determinants to identify the one that has an inverse. Then find the inverse. A=[141020101]B=[140020301]12T 13T 14T Find the maximum and minimum values of the given objective function on the indicated feasible region. 1. M = 200 x y 2P 3P Find the maximum and minimum value of the given objective function on the indicated feasible region. 4. Q = 70x + 82y {x0,y0x10,y20x+y5x+2y18 Making Furniture A furniture manufacturer makes wooden tables and chairs. The production process involves two basic types of labor: carpentry and finishing. A table requires 2 h of carpentry and 1 h of finishing, and a chair requires 3 h of carpentry and 12h of finishing. The profit is 35 per table and 20 per chair. The manufacturers employees can supply a maximum of 108 h of carpentry work and 20 h of finishing work per day. How many tables and chairs should be made each day to maximize profit?A Housing Development A housing contractor has subdivided a farm into 100 building lots. She has designed two types of homes for these lots: colonial and ranch style. A colonial requires 30,000 of capital and produces a profit of 4000 when sold. A ranch-style house requires 40,000 of capital and provides an 8000 profit. If the contractor has 3.6 million of capital on hand, how many houses of each type should she build for maximum profit? Will any of the lots be left vacant?Hauling Fruit A trucker hauls citrus fruit from Florida to Montreal. Each crate of oranges is 4 ft3 in volume and weighs 80 lb. Each crate of grapefruit has a volume of 6 ft3 and weighs 100 lb. His truck has a maximum capacity of 300 ft3 and can carry no more than 5600 lb. Moreover, he is not permitted to carry more crates of grapefruit than crates of oranges. If his profit is 2.50 on each crate of oranges and 4 on each crate of grapefruit, how many crates of each fruit should he carry for maximum profit?Manufacturing Calculators A manufacturer of calculators produces two models: standard and scientific. Long-term demand for the two models mandates that the company manufacture at least 100 standard and 80 scientific calculators each day. However, because of limitations on production capacity, no more than 200 standard and 170 scientific calculators can be made daily. To satisfy a shipping contract, a total of at least 200 calculators must be shipped every day. (a) If the production cost is 5 for a standard calculator and 7 for a scientific one, how many of each model should be produced daily to minimize this cost? (b) If each standard calculator results in a 2 loss but each scientific one produces a 5 profit, how many of each model should be made daily to maximize profit?Shipping Televisions An electronics discount chain has a sale on a certain brand of 60-in. high-definition television set. The chain has stores in Santa Monica and El Toro and warehouses in Long Beach and Pasadena. To satisfy rush orders, 15 sets must be shipped from the warehouses to the Santa Monica store, and 19 must be shipped to the El Toro store. The cost of shipping a set is 5 from Long Beach to Santa Monica, 6 from Long Beach to El Toro, 4 from Pasadena to Santa Monica, and 5.50 from Pasadena to El Toro. If The Long Beach warehouse has 24 sets and the Pasadena warehouse has 18 sets in stock, how many sets should be shipped from each warehouse to each store to fill the orders at a minimum shipping cost?10P Packaging Nuts A confectioner sells two types of nut mixtures. The standard-mixture package contains 100 g of cashews and 200 g of peanuts and sells for 1.95. The deluxe-mixture package contains 150 g of cashews and 50 g of peanuts and sells for 2.25. The confectioner has 15 kg of cashews and 20 kg of peanuts available. On the basis of past sales, the confectioner needs to have at least as many standard as deluxe packages available. How many bags of each mixture should he package to maximize his revenue?Feeding Lab Rabbits A biologist wishes to feed laboratory rabbits a mixture of two types of foods. Type I contains 8 g of fat, 12 g of carbohydrate, and 2 g of protein per ounce. Type II contains 12 g of fat, 12 g of carbohydrate, and 1 g of protein per ounce. Type I costs 0.20 per ounce and type II costs 0.30 per ounce. Each rabbit receives a daily minimum of 24 g of fat 36 g of carbohydrate, and 4 g of protein, but get no more than 5 oz of food per day. How many ounces of each food type should be fed to each rabbit daily to satisfy the dietary requirements at minimum cost?Investing in Bonds A woman wishes to invest 12,000 in three types of bonds: municipal bonds paying 7% interest per year, bank certificates paying 8%, and high-risk bonds paying 12%. For tax reasons she wants the amount invested in municipal bonds to be at least three times the amount invested in bunk certificates. To keep her level of risk manage-able, she will invest no more than 2000 in high-risk bonds. How much should she invest in each type of bond to maximize her annual interest yield? [Hint: Let x = amount in municipal bonds and y = amount in bunk certificates. Then the amount in high-risk bonds will be 12,000 x y.]14P Business Strategy A small software company publishes computer games, educational software, and utility software. Their business strategy is to market a total of 36 new programs each year, at least four of these being games. The number of utility programs published is never more than twice the number of educational programs. On average, the company makes an annual profit of 5000 on each computer game, 8000 on each educational program, and 6000 on each utility program. How many of each type of software should the company publish annually for maximum profit?Feasible Region All parts of this problem refer to the following feasible region and objective function. {x0xyx+2y12x+y10P=x+4y (a) Graph the feasible region. (b) On your graph from part (a), sketch the graphs of the linear equations obtained by setting P equal to 40, 36, 32, and 28. (c) If you continue to decrease the value of P, at which vertex of the feasible region will these lines first touch the feasible region? (d) Verify that the maximum value of P on the feasible region occurs at the vertex you chose in part (c).A parabola is the set of all points in the plane that are equidistant from a fixed point called the __________ and a fixed line called the __________ of the parabola.The graph of the equation x2 = 4py is a parabola with focus F(___, ___) and directrix y = __________. So the graph of x2 = 12y is a parabola with focus F(___, ___) and directrix y = _______________ .The graph of the equation y2 = 4px is a parabola with focus F(___, ___) and directrix x = __________. So the graph of y2 = 12x is a parabola with focus F(___, ___) and directrix x = __________.Label the focus, directrix, and vertex on the graphs given for the parabolas in Exercises 2 and 3. (a) x2 = 12y (b) y2 = 12x Graphs of Parabolas Match the equation with the graphs labeled IVI. Give reasons for your answers. 5. y2 = 2x 6E Graphs of Parabolas Match the equation with the graphs labeled IVI. Give reasons for your answers. 7. x2 = 6y 8E Graphs of Parabolas Match the equation with the graphs labeled IVI. Give reasons for your answers. 9. y2 8x = 0 Graphs of Parabolas Match the equation with the graphs labeled IVI. Give reasons for your answers. 10. 12y + x2 = 0 Graphing Parabolas An equation of a parabola is given. (a) Find the focus, directrix, and focal diameter of the parabola. (b) Sketch a graph of the parabola and its directrix. 11. x2 = 8y 12E Graphing Parabolas An equation of a parabola is given. (a) Find the focus, directrix, and focal diameter of the parabola. (b) Sketch a graph of the parabola and its directrix. 13. y2 = 24x 14E 15E Graphing Parabolas An equation of a parabola is given. (a) Find the focus, directrix, and focal diameter of the parabola. (b) Sketch a graph of the parabola and its directrix. 16. x = 2y2 17E 18E Graphing Parabolas An equation of a parabola is given. (a) Find the focus, directrix, and focal diameter of the parabola. (b) Sketch a graph of the parabola and its directrix. 19. 5y = x2 20E 21E 22E Graphing Parabolas An equation of a parabola is given. (a) Find the focus, directrix, and focal diameter of the parabola. (b) Sketch a graph of the parabola and its directrix. 23. 5x + 3y2 = 0 24E 25E 26E 27E 28E 29E 30E Finding the Equation of a Parabola Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). 31. Focus: F(0, 6)32E 33E 34E 35E 36E Finding the Equation of a Parabola Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). 37. Directrix: x = 4 Finding the Equation of a Parabola Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). 38. Directrix: y=12 Finding the Equation of a Parabola Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). 39. Directrix: y=110 40E 41E 42E Finding the Equation of a Parabola Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). 43. Focus on the positive x-axis, 2 units away from the directrix Finding the Equation of a Parabola Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). 44. Focus on the negative y-axis, 6 units away from the directrix Finding the Equation of a Parabola Find an equation for the parabola that has its vertex at the origin and satisfies the given condition(s). 45. Opens downward with focus 10 units away from the vertex 46E 47E 48E Finding the Equation of a Parabola Find an equation of the parabola whose graph is shown. 49.50E Finding the Equation of a Parabola Find an equation of the parabola whose graph is shown. 51.Finding the Equation of a Parabola Find an equation of the parabola whose graph is shown. 52.Finding the Equation of a Parabola Find an equation of the parabola whose graph is shown. 53.54E Finding the Equation of a Parabola Find an equation of the parabola whose graph is shown. 55.56E 57E Finding the Equation of a Parabola Find an equation of the parabola whose graph is shown. 58.59E Families of Parabolas (a) Find equations for the family of parabolas with the given description. (b) Draw the graphs. What do you conclude? 60. The family of parabolas with vertex at the origin, focus on the positive y-axis, and with focal diameters 1, 2, 4, and 8 Parabolic Reflector A lamp with a parabolic reflector is shown in the figure. The bulb is placed at the focus, and the focal diameter is 12 cm. (a) Find an equation of the parabola. (b) Find the diameter d(C, D) of the opening, 20 cm from the vertex.Satellite Dish A reflector for a satellite dish is parabolic in cross section, with the receiver at the focus F. The reflector is 1 ft deep and 20 ft wide from rim to rim (see the figure). How far is the receiver from the vertex of the parabolic reflector?Suspension Bridge In a suspension bridge the shape of the suspension cables is parabolic. The bridge shown in the figure has towers that are 600 m apart, and the lowest point of the suspension cables is 150 m below the top of the towers. Find the equation of the parabolic part of the cables, placing the origin of the coordinate system at the vertex. [Note: This equation is used to find the length of cable needed in the construction of the bridge.]Reflecting Telescope The Hale telescope at the Mount Palomar Observatory has a 200-in. mirror, as shown in the figure. The mirror is constructed in a parabolic shape that collects light from the stars and focuses it at the prime focus, that is, the focus of the parabola. The mirror is 3.79 in. deep at its center. Find the focal length of this parabolic mirror, that is, the distance from the vertex to the focus.65E 66E An ellipse is the set of all points in the plane for which the __________ of the distances from two fixed points F1 and F2 is constant. The points F1, and F2 are called the __________ of the ellipse.The graph of the equation x2a2+y2b2=1 with a b 0 is an ellipse with vertices (_____, _____ ) and (_____, _____) and foci ( c, 0), where c = __________. So the graph of x252+y242=1 is an ellipse with vertices (_____, _____ ) and (_____, _____ ) and foci (_____, _____) and (_____, _____).The graph of the equation x2b2+y2a2=1 with a b 0 is an ellipse with vertices (_____, _____ ) and (_____, _____) and foci (0, c), where c = __________. So the graph of x242+y252=1is an ellipse with vertices (_____, _____) and (_____, _____) and foci (_____, _____ ) and (_____, _____).Label the vertices and foci on the graphs given for the ellipses in Exercises 2 and 3. (a) x252+y242=1 (b) x242+y252=1 Graphs of Ellipses Match the equation with the graphs labeled IIV. Give reasons for your answers. 5. x216+y24=1 6E Graphs of Ellipses Match the equation with the graphs labeled IIV. Give reasons for your answers. 7. 4x2 + y2 = 4 8E Graphing Ellipses An equation of an ellipse is given. (a) Find the vertices, foci, and eccentricity of the ellipse. (b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse. 9. x225+y29=1 10E 11E Graphing Ellipses An equation of an ellipse is given. (a) Find the vertices, foci, and eccentricity of the ellipse. (b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse. 12. x24+y2=1 13E Graphing Ellipses An equation of an ellipse is given. (a) Find the vertices, foci, and eccentricity of the ellipse. (b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse. 14. x29+y264=1 15E 16E Graphing Ellipses An equation of an ellipse is given. (a) Find the vertices, foci, and eccentricity of the ellipse. (b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse. 17. x2 + 4y2 = 16 18E Graphing Ellipses An equation of an ellipse is given. (a) Find the vertices, foci, and eccentricity of the ellipse. (b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse. 19. 16x2 + 25y2 = 1600 Graphing Ellipses An equation of an ellipse is given. (a) Find the vertices, foci, and eccentricity of the ellipse. (b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse. 20. 2x2 + 49y2 = 98 21E 22E 23E 24E 25E 26E 27E 28E 29E Finding the Equation of an Ellipse Find an equation for the ellipse whose graph is shown. 30.Finding the Equation of an Ellipse Find an equation for the ellipse whose graph is shown. 31.32E Finding the Equation of an Ellipse Find an equation for the ellipse whose graph is shown. 33.34E 35E 36E 37E 38E 39E 40E Finding the Equation of an Ellipse Find an equation for the ellipse that satisfies the given conditions. 41. Foci: F(1, 0), vertices: (2, 0)42E 43E 44E 45E Finding the Equation of an Ellipse Find an equation for the ellipse that satisfies the given conditions. 46. Length of major axis: 6, length of minor axis: 4, foci on x-axis Finding the Equation of an Ellipse Find an equation for the ellipse that satisfies the given conditions. 47. Foci: (0, 2), length of minor axis: 6 48E 49E Finding the Equation of an Ellipse Find an equation for the ellipse that satisfies the given conditions. 50. Endpoints of minor axis: (0, 3), distance between foci: 8 51E Finding the Equation of an Ellipse Find an equation for the ellipse that satisfies the given conditions. 52. Length of minor axis: 10, foci on y-axis, ellipse passes through the point (5,40)Finding the Equation of an Ellipse Find an equation for the ellipse that satisfies the given conditions. 53. Eccentricity: 13, foci: (0, 2)Finding the Equation of an Ellipse Find an equation for the ellipse that satisfies the given conditions. 54. Eccentricity: 0.75, foci: (1.5, 0)55E 56E 57E Intersecting Ellipses Find the intersection points of the pair of ellipses. Sketch the graphs of each pair of equations on the same coordinate axes, and label the points of intersection. 58. {x216+y29=1x29+y216=1 59E 60E Ancillary Circle The ancillary circle of an ellipse is the circle with radius equal to half the length of the minor axis and center the same as the ellipse (see the figure). The ancillary circle is thus the largest circle that can fit within an ellipse. (a) Find an equation for the ancillary circle of the ellipse x2 + 4y2 = 16. (b) For the ellipse and ancillary circle of part (a), show that if (s, t) is a point on the ancillary circle, then (2s, t) is a point on the ellipse.62E Family of Ellipses If k 0, the following equation represents an ellipse: x2k+y24+k=1 Show that all the ellipses represented by this equation have the same foci, no matter what the value of k.How Wide Is an Ellipse at a Focus? A latus rectum for an ellipse is a line segment perpendicular to the major axis at a focus, with endpoints on the ellipse, as shown in the figure. Show that the length of a latus rectum is 2b2/a for the ellipse x2a2+y2b2=1ab Perihelion and Aphelion The planets move around the sun in elliptical orbits with the sun at one focus. The point in the orbit at which the planet is closest to the sun is called perihelion, and the point at which it is farthest is called aphelion. These points are the vertices of the orbit. The earths distance from the sun is 147,000,000 km at perihelion and 153,000,000 km at aphelion. Find an equation for the earths orbit. (Place the origin at the center of the orbit with the sun on the x-axis.)The Orbit of Pluto With an eccentricity of 0.25, Plutos orbit is the most eccentric in the solar system. The length of the minor axis of its orbit is approximately 10,000,000,000 km. Find the distance between Pluto and the sun at perihelion and at aphelion. (See Exercise 65.)Lunar Orbit For an object in an elliptical orbit around the moon, the points in the orbit that are closest to and farthest from the center of the moon arc called perilune and apolune, respectively. These are the vertices of the orbit. The center of the moon is at one focus of the orbit. The Apollo 11 spacecraft was placed in a lunar orbit with perilune at 68 mi and apolune at 195 mi above the surface of the moon. Assuming that the moon is a sphere of radius 1075 mi, find an equation for the orbit of Apollo 11. (Place the coordinate axes so that the origin is at the center of the orbit and the foci are located on the x-axis.)Plywood Ellipse A carpenter wishes to construct an elliptical table top from a 4 ft by 8 ft sheet of plywood. He will trace out the ellipse using the thumbtack and string method illustrated in Figures 2 and 3. What length of string should he use, and how far apart should the tacks be located, if the ellipse is to be the largest possible that can be cut out of the plywood sheet?Sunburst Window A sunburst window above a doorway is constructed in the shape of the top half of an ellipse, as shown in the figure. The window is 20 in. tall at its highest point and 80 in. wide at the bottom. Find the height of the window 25 in. from the center of the base.DISCUSS: Drawing an Ellipse on a Blackboard Try drawing an ellipse as accurately as possible on a blackboard. How would a piece of string and two friends help this process?DISCUSS: Light Cone from a Flashlight A flashlight shines on a wall, as shown in the figure. What is the shape of the boundary of the lighted area? Explain your answer.DISCUSS: Is It an Ellipse? A piece of paper is wrapped around a cylindrical bottle, and then a compass is used to draw a circle on the paper, as shown in the figure. When the paper is laid flat, is the shape drawn on the paper an ellipse? (You dont need to prove your answer, but you might want to do the experiment and see what you get.)A hyperbola is the set of all points in the plane for which the __________ of the distances from two fixed points F1 and F2 is constant. The points F1 and F2 are called the __________ of the hyperbola.The graph of the equation x2a2y2b2=1 with a 0, b 0 is a hyperbola with __________ (horizontal/vertical) transverse axis, vertices (___, ___) and (___, ___) and foci (c, 0), where c = __________. So the graph of x242y232=1 is a hyperbola with vertices (___, ___) and (___, ___) and foci (___, ___) and (___, ___).3E Label the vertices, foci, and asymptotes on the graphs given for the hyperbolas in Exercises 2 and 3. (a) x242y232=1 (b) y242x232=1 Graphs of Hyperbolas Match the equation with the graphs labeled IIV. Give reasons for your answers. 5. x24y2=1 6E Graphs of Hyperbolas Match the equation with the graphs labeled IIV. Give reasons for your answers. 7. 16y2 x2 = 144 8E Graphing Hyperbolas An equation of a hyperbola is given. (a) Find the vertices, foci, and asymptotes of the hyperbola. (b) Determine the length of the transverse axis. (c) Sketch a graph of the hyperbola. 9. x24y216=1 10E Graphing Hyperbolas An equation of a hyperbola is given. (a) Find the vertices, foci, and asymptotes of the hyperbola. (b) Determine the length of the transverse axis. (c) Sketch a graph of the hyperbola. 11. y236x24=1 12E 13E 14E 15E 16E 17E 18E Graphing Hyperbolas An equation of a hyperbola is given. (a) Find the vertices, foci, and asymptotes of the hyperbola. (b) Determine the length of the transverse axis. (c) Sketch a graph of the hyperbola. 19. 4y2 9x2 = 144 20E 21E 22E 23E 24E 25E 26E Finding an Equation of a Hyperbola Find the equation for the hyperbola whose graph is shown. 27.28E 29E 30E 31E Finding an Equation of a Hyperbola Find the equation for the hyperbola whose graph is shown. 32.33E 34E 35E 36E Finding the Equation of a Hyperbola Find an equation for the hyperbola that satisfies the given conditions. 37. Foci: (5, 0), vertices: (3, 0)38E Finding the Equation of a Hyperbola Find an equation for the hyperbola that satisfies the given conditions. 39. Foci: (0, 2), vertices: (0, 1)40E Finding the Equation of a Hyperbola Find an equation for the hyperbola that satisfies the given conditions. 41. Vertices: (1, 0), asymptotes: y = 5x 42E 43E Finding the Equation of a Hyperbola Find an equation for the hyperbola that satisfies the given conditions. 44. Vertices: (2, 0), hyperbola passes through (3,30)Finding the Equation of a Hyperbola Find an equation for the hyperbola that satisfies the given conditions. 45. Asymptotes: y = x, hyperbola passes through (5, 3)Finding the Equation of a Hyperbola Find an equation for the hyperbola that satisfies the given conditions. 46. Asymptotes: y = x, hyperbola passes through (1, 2)Finding the Equation of a Hyperbola Find an equation for the hyperbola that satisfies the given conditions. 47. Foci: (0, 3), hyperbola passes through (1, 4)48E 49E 50E Perpendicular Asymptotes (a) Show that the asymptotes of the hyperbola x2 y2 = 5 are perpendicular to each other. (b) Find an equation for the hyperbola with foci (c, 0) and with asymptotes perpendicular to each other.52E 53E 54E 55E Navigation In the figure on the next page, the LORAN stations at A and B are 500 mi apart, and the ship at P receives station As signal 2640 microseconds (s) before it receives the signal from station B. (a) Assuming that radio signals travel at 980 ft/s, find d(P, A) d(P, B). (b) Find an equation for the branch of the hyperbola indicated in red in the figure. (Use miles as the unit of distance.) (c) If A is due north of B and if P is due east of A, how far is P from A?Comet Trajectories Some comets, such as Halleys comet, are a permanent part of the solar system, traveling in elliptical orbits around the sun. Other comets pass through the solar system only once, following a hyperbolic path with the sun at a focus. The figure below shows the path of such a comet. Find an equation for the path, assuming that the closest the comet comes to the sun is 2 109 mi and that the path the comet was taking before it neared the solar system is at a right angle to the path it continues on after leaving the solar system.Ripples in Pool Two stones are dropped simultaneously into a calm pool of water. The crests of the resulting waves form equally spaced concentric circles, as shown in the figures. The waves interact with each other to create certain interference patterns. (a) Explain why the red dots lie on an ellipse. (b) Explain why the blue dots lie on a hyperbola.59E DISCUSS: Light from a Lamp The light from a lamp forms a lighted area on a wall, as shown in the figure. Why is the boundary of this lighted area a hyperbola? How can one hold a flashlight so that its beam forms a hyperbola on the ground?1E The graphs of x2 = 12y and (x 3)2 = 12(y 1) are given. Label the focus, directrix, and vertex on each parabola.3E 4E Graphing Shifted Ellipses An equation of an ellipse is given. (a) Find the center, vertices, and foci of the ellipse. (b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse. 5. (x2)29+(y1)24=1 6E 7E 8E 9E 10E Graphing Shifted Ellipses An equation of an ellipse is given. (a) Find the center, vertices, and foci of the ellipse. (b) Determine the lengths of the major and minor axes. (c) Sketch a graph of the ellipse. 11. 4x2 + 25y 50y = 75 12E Graphing Shifted Parabolas An equation of a parabola is given. (a) Find the vertex, focus, and directrix of the parabola. (b) Sketch a graph showing the parabola and its directrix. 13. (x 3)2 = 8(y + 1)14E 15E 16E 17E 18E Graphing Shifted Parabolas An equation of a parabola is given. (a) Find the vertex, focus, and directrix of the parabola. (b) Sketch a graph showing the parabola and its directrix. 19. y2 6y 12x + 33 = 0 20E Graphing Shifted Hyperbolas An equation of a hyperbola is given. (a) Find the center, vertices, foci, and asymptotes of the hyperbola. (b) Sketch a graph showing the hyperbola and its asymptotes. 21. (x+1)29(y3)216=1 22E 23E 24E 25E 26E Graphing Shifted Hyperbolas An equation of a hyperbola is given. (a) Find the center, vertices, foci, and asymptotes of the hyperbola. (b) Sketch a graph showing the hyperbola and its asymptotes. 27. 36x2 + 72x 4y2 + 32y + 116 = 0 28E 29E 30E 31E 32E 33E 34E 35E Finding the Equation of a Shifted Conic Find an equation for the conic section with the given properties. 36. The ellipse with vertices V1(1, 4) and V2(1, 6) and foci F1(1, 3) and F2(1, 5)37E 38E 39E 40E 41E 42E 43E 44E Finding the Equation of a Shifted Conic Find an equation for the conic section with the given properties. 45. The parabola that passes through the point (6, 1), with vertex V(1, 2) and horizontal axis of symmetry 46E 47E 48E Graphing Shifted Conics Complete the square to determine whether the graph of the equation is an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. 49. x2 5y2 2x + 20y = 44 50E Graphing Shifted Conics Complete the square to determine whether the graph of the equation is an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. 51. 4x2 + 25y2 24x + 250y + 561 = 0 52E Graphing Shifted Conics Complete the square to determine whether the graph of the equation is an ellipse, a parabola, a hyperbola, or a degenerate conic. If the graph is an ellipse, find the center, foci, vertices, and lengths of the major and minor axes. If it is a parabola, find the vertex, focus, and directrix. If it is a hyperbola, find the center, foci, vertices, and asymptotes. Then sketch the graph of the equation. If the equation has no graph, explain why. 53. 16x2 9y2 96x + 288 = 0

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