Bartleby Sitemap - Textbook Solutions
All Textbook Solutions for Precalculus Enhanced with Graphing Utilities
In Problems 21-38, find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand. Vertex at ( 0,0 ) ; axis of symmetry the y-axis ; containing the point ( 2,3 )In Problems 21-38, find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand. Vertex at ( 0,0 ) ; axis of symmetry the x-axis ; containing the point ( 2,3 )In Problems 21-38, find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand. Vertex at ( 2,3 ) ; focus at ( 2,5 )In Problems 21-38, find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand. Vertex at ( 4,2 ) ; focus at ( 6,2 )In Problems 21-38, find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand. Vertex at ( 1,2 ) ; focus at ( 0,2 )In Problems 21-38, find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand. Vertex at ( 3,0 ) ; focus at ( 3,2 )In Problems 21-38, find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand. Vertex at ( 3,4 ) ; directrix the line y=2In Problems 21-38, find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand. Vertex at ( 2,4 ) ; directrix the line x=4In Problems 21-38, find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand. Focus at ( 3,2 ) ; directrix the line x=1In Problems 21-38, find the equation of the parabola described. Find the two points that define the latus rectum, and graph the equation by hand. Focus at ( 4,4 ) ; directrix the line y=2In Problems 39-56, find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility. x 2 =4yIn Problems 39-56, find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility. y 2 =8xIn Problems 39-56, find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility. y 2 =16xIn Problems 39-56, find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility. x 2 =4yIn Problems 39-56, find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility. ( y2 ) 2 =8( x+1 )In Problems 39-56, find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility. ( x+4 ) 2 =16( y+2 )In Problems 39-56, find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility. ( x3 ) 2 =( y+1 )In Problems 39-56, find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility. ( y+1 ) 2 =4( x2 )In Problems 39-56, find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility. ( y+3 ) 2 =8( x2 )In Problems 39-56, find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility. ( x2 ) 2 =4( y3 )In Problems 39-56, find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility. y 2 4y+4x+4=0In Problems 39-56, find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility. x 2 +6x4y+1=0In Problems 39-56, find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility. x 2 +8x=4y8In Problems 39-56, find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility. y 2 2y=8x1In Problems 39-56, find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility. y 2 +2yx=0In Problems 39-56, find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility. x 2 4x=2yIn Problems 39-56, find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility. x 2 4x=y+4In Problems 39-56, find the vertex, focus, and directrix of each parabola. Graph the equation by hand. Verify your graph using a graphing utility. y 2 +12y=x+1In Problems 57-64, write an equation for each parabola.In Problems 57-64, write an equation for each parabola.In Problems 57-64, write an equation for each parabola.In Problems 57-64, write an equation for each parabola.In Problems 57-64, write an equation for each parabola.In Problems 57-64, write an equation for each parabola.In Problems 57-64, write an equation for each parabola.In Problems 57-64, write an equation for each parabola.Satellite Dish A satellite dish is shaped like a paraboloid of revolution. The signals that emanate from a satellite strike the surface of the dish and are reflected to a single point, where the receiver is located. If the dish is 10 feet across at its opening and 4 feet deep at its center, at what position should the receiver be placed?Constructing a TV Dish A cable TV receiving dish is in the shape of a paraboloid of revolution. Find the location of the receiver, which is placed at the focus, if the dish is 6 feet across at its opening and 2 feet deep.Constructing a Flashlight The reflector of a flashlight is in the shape of a paraboloid of revolution. Its diameter is 4 inches and its depth is 1 inch. How far from the vertex should the light bulb be placed so that the rays will be reflected parallel to the axis?Constructing a Headlight A sealed-beam headlight is in the shape of a paraboloid of revolution. The bulb, which is placed at the focus, is 1 inch from the vertex. If the depth is to be 2 inches, what is the diameter of the headlight at its opening?Suspension Bridge The cables of a suspension bridge are in the shape of a parabola, as shown in the figure. The towers supporting the cable are 600 feet apart and 80 feet high. If the cables touch the road surface midway between the towers, what is the height of the cable from the road at a point 150 feet from the center of the bridge?Suspension Bridge The cables of a suspension bridge are in the shape of a parabola. The towers supporting the cable are 400 feet apart and 100 feet high. If the cables are at a height of 10 feet midway between the towers, what is the height of the cable at a point 50 feet from the center of the bridge?Searchlight A searchlight is shaped like a paraboloid of revolution. If the light source is located 2 feet from the base along the axis of symmetry and the opening is 5 feet across, how deep should the searchlight be?Searchlight A searchlight is shaped like a paraboloid of revolution. If the light source is located 2 feet from the base along the axis of symmetry and the depth of the searchlight is 4 feet, what should the width of the opening be?Solar Heat A mirror is shaped like a paraboloid of revolution and will be used to concentrate the rays of the sun at its focus, creating a heat source. See the figure. If the mirror is 20 feet across at its opening and is 6 feet deep, where will the heat source be concentrated?Reflecting Telescope A reflecting telescope contains a mirror shaped like a paraboloid of revolution. If the mirror is 4 inches across at its opening and is 3 inches deep, where will the collected light be concentrated?Parabolic Arch Bridge A bridge is built in the shape of a parabolic arch. The bridge has a span of 120 feet and a maximum height of 25 feet. See the illustration. Choose a suitable rectangular coordinate system and find the height of the arch at distances of 10,30, and 50 feet from the center.Parabolic Arch Bridge A bridge is to be built in the shape of a parabolic arch and is to have a span of 100 feet. The height of the arch a distance of 40 feet from the center is to be 10 feet. Find the height of the arch at its center.Gateway Arch The Gateway Arch in St. Louis is often mistaken to be parabolic in shape. In fact, it is a catenary, which has a more complicated formula than a parabola. The Arch is 630 feet high and 630 feet wide at its base.Show that an equation of the form A x 2 +Ey=0 A0 , E0 is the equation of a parabola with vertex at ( 0,0 ) and axis of symmetry the y-axis . Find its focus and directrix.Show that an equation of the form C y 2 +Dx=0 C0 , D0 is the equation of a parabola with vertex at ( 0,0 ) and axis of symmetry the x-axis . Find its focus and directrix.Show that the graph of an equation of the form A x 2 +Dx+Ey+F=0 A0 (a) Is a parabola if E0 . (b) Is a vertical line if E=0 and D 2 4AF=0 . (c) Is two vertical lines if E=0 and D 2 4AF0 . (d) Contains no points if E=0 and D 2 4AF0 .Show that the graph of an equation of the form C y 2 +Dx+Ey+F=0 C0 (a) Is a parabola if D0 . (b) Is a horizontal line if D=0 and E 2 4CF=0 . (c) Is two horizontal lines if D=0 and E 2 4CF0 . (d) Contains no points if D=0 and E 2 4CF0 .The distance d from P 1 =( 2,5 ) to P 2 =( 4,2 ) is d= ______. (p.4)To complete the square of x 2 3x , Add _____. (p. A28-A29)Find the intercepts of the equation y 2 =164 x 2 . (pp. 18-19)The point that is symmetric with respect to the y-axis to the point ( 2,5 ) is ________. (pp. 19-21)The point that is symmetric with respect to the y-axis to the point ( 2,5 ) is _______. (pp. 19-21)6AYUA(n) _______ is the collection of all points in a plane the sum of whose distances from two fixed points is a constant.For an ellipse, the foci lie on a line called the ______. (a) minor axis (b) major axis (c) directrix (d) latus rectumFor the ellipse x 2 4 + y 2 25 =1 , the vertices are the points _______ and _______.For the ellipse x 2 25 + y 2 9 =1 , the value of a is ______, the value of b is ______, and the major axis is the ______ -axis.If the center of an ellipse is ( 2,3 ) , the major axis is parallel to the x-axis , and the distance from the center of the ellipse to its vertices is a=4 units, then the coordinates of the vertices are _______ and _______.If the foci of an ellipse are ( 4,4 ) and ( 6,4 ) , then the coordinates of the center of the ellipse are _________. (a) ( 1,4 ) (b) ( 4,1 ) (c) ( 1,0 ) (d) ( 5,4 )In problems 13-16, the graph of an ellipse is given. Match each graph to its equation. (A) x 2 4 + y 2 =1 (B) x 2 + y 2 4 =1 (C) x 2 16 + y 2 4 =1 (D) x 2 4 + y 2 16 =1In problems 13-16, the graph of an ellipse is given. Match each graph to its equation. (A) x 2 4 + y 2 =1 (B) x 2 + y 2 4 =1 (C) x 2 16 + y 2 4 =1 (D) x 2 4 + y 2 16 =1In problems 13-16, the graph of an ellipse is given. Match each graph to its equation. (A) x 2 4 + y 2 =1 (B) x 2 + y 2 4 =1 (C) x 2 16 + y 2 4 =1 (D) x 2 4 + y 2 16 =1In problems 13-16, the graph of an ellipse is given. Match each graph to its equation. (A) x 2 4 + y 2 =1 (B) x 2 + y 2 4 =1 (C) x 2 16 + y 2 4 =1 (D) x 2 4 + y 2 16 =1In problems 17-26, find the vertices and foci of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility. x 2 25 + y 2 4 =1In problems 17-26, find the vertices and foci of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility. x 2 9 + y 2 4 =1In problems 17-26, find the vertices and foci of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility. x 2 9 + y 2 25 =1In problems 17-26, find the vertices and foci of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility. x 2 + y 2 16 =1In problems 17-26, find the vertices and foci of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility. 4 x 2 + y 2 =16In problems 17-26, find the vertices and foci of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility. x 2 +9 y 2 =18In problems 17-26, find the vertices and foci of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility. 4 y 2 + x 2 =8In problems 17-26, find the vertices and foci of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility. 4 y 2 +9 x 2 =36In problems 17-26, find the vertices and foci of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility. x 2 + y 2 =16In problems 17-26, find the vertices and foci of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility. x 2 + y 2 =4In Problems 27-38, find an equation for each ellipse. Graph the equation by hand. Center at ( 0,0 ) ; focus at ( 3,0 ) ; vertex at ( 5,0 )In Problems 27-38, find an equation for each ellipse. Graph the equation by hand. Center at ( 0,0 ) ; focus at ( 1,0 ) ; vertex at ( 3,0 )In Problems 27-38, find an equation for each ellipse. Graph the equation by hand. Center at ( 0,0 ) ; focus at ( 0,4 ) ; vertex at ( 0,5 )In Problems 27-38, find an equation for each ellipse. Graph the equation by hand. Center at ( 0,0 ) ; focus at ( 0,1 ) ; vertex at ( 0,2 )In Problems 27-38, find an equation for each ellipse. Graph the equation by hand. Foci at ( 2,0 ) ; length of the major axis is 6In Problems 27-38, find an equation for each ellipse. Graph the equation by hand. Foci at ( 0,2 ) ; length of the major axis is 8In Problems 27-38, find an equation for each ellipse. Graph the equation by hand. Focus at ( 4,0 ) ; vertices at ( 5,0 )In Problems 27-38, find an equation for each ellipse. Graph the equation by hand. Focus at ( 0,4 ) ; vertices at ( 0,8 )In Problems 27-38, find an equation for each ellipse. Graph the equation by hand. Foci at ( 0,3 ) ; x-intercepts are 2In Problems 27-38, find an equation for each ellipse. Graph the equation by hand. Vertices at ( 4,0 ) ; y-intercepts are 1In Problems 27-38, find an equation for each ellipse. Graph the equation by hand. Center at ( 0,0 ) ; vertex at ( 0,4 ) ; b=1In Problems 27-38, find an equation for each ellipse. Graph the equation by hand. Vertices at ( 5,0 ) ; c=2In Problems 39-42, write an equation for each ellipse.In Problems 39-42, write an equation for each ellipse.In Problems 39-42, write an equation for each ellipse.In Problems 39-42, write an equation for each ellipse.In Problems 43-54, analyze each equation; that is, find the center, foci, and vertices of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility. ( x3 ) 2 4 + ( y+1 ) 2 9 =1In Problems 43-54, analyze each equation; that is, find the center, foci, and vertices of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility. ( x+4 ) 2 9 + ( y+2 ) 2 4 =1In Problems 43-54, analyze each equation; that is, find the center, foci, and vertices of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility. ( x+5 ) 2 +4 ( y4 ) 2 =16In Problems 43-54, analyze each equation; that is, find the center, foci, and vertices of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility. 9 ( x3 ) 2 + ( y+2 ) 2 =18In Problems 43-54, analyze each equation; that is, find the center, foci, and vertices of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility. x 2 +4x+4 y 2 8y+4=0In Problems 43-54, analyze each equation; that is, find the center, foci, and vertices of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility. x 2 +3 y 2 12y+9=0In Problems 43-54, analyze each equation; that is, find the center, foci, and vertices of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility. 2 x 2 +3 y 2 8x+6y+5=0In Problems 43-54, analyze each equation; that is, find the center, foci, and vertices of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility. 4 x 2 +3 y 2 +8x6y=5In Problems 43-54, analyze each equation; that is, find the center, foci, and vertices of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility. 9 x 2 +4 y 2 18x+16y11=0In Problems 43-54, analyze each equation; that is, find the center, foci, and vertices of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility. x 2 +9 y 2 +6x18y+9=0In Problems 43-54, analyze each equation; that is, find the center, foci, and vertices of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility. 4 x 2 + y 2 +4y=0In Problems 43-54, analyze each equation; that is, find the center, foci, and vertices of each ellipse. Graph each equation by hand. Verify your graph using a graphing utility. 9 x 2 + y 2 18x=0In Problems 55-64, find an equation for each ellipse. Graph the equation by hand. Center at ( 2,2 ) ; vertex at ( 7,2 ) ; focus at ( 4,2 )In Problems 55-64, find an equation for each ellipse. Graph the equation by hand. Center at ( 3,1 ) ; vertex at ( 3,3 ) ; focus at ( 3,0 )In Problems 55-64, find an equation for each ellipse. Graph the equation by hand. Center at ( 4,3 ) ; vertex at ( 4,9 ) ; focus at ( 4,8 )In Problems 55-64, find an equation for each ellipse. Graph the equation by hand. Foci at ( 1,2 ) and ( 3,2 ) ; vertex at ( 4,2 )In Problems 55-64, find an equation for each ellipse. Graph the equation by hand. Foci at ( 5,1 ) and ( 1,1 ) ; length of the major axis is 8In Problems 55-64, find an equation for each ellipse. Graph the equation by hand. Vertices at ( 2,5 ) and ( 2,1 ) ; c=2In Problems 55-64, find an equation for each ellipse. Graph the equation by hand. Center at ( 1,2 ) ; focus at ( 4,2 ) ; contains the point ( 1,3 )In Problems 55-64, find an equation for each ellipse. Graph the equation by hand. Center at ( 1,2 ) ; focus at ( 1,4 ) ; contains the point ( 2,2 )In Problems 55-64, find an equation for each ellipse. Graph the equation by hand. Center at ( 1,2 ) ; vertex at ( 4,2 ) ; contains the point ( 1,5 )In Problems 55-64, find an equation for each ellipse. Graph the equation by hand. Center at ( 1,2 ) ; vertex at ( 1,4 ) ; contains the point ( 1+ 3 ,3 )In Problems 65-68, graph each function. Be sure to label all the intercepts. [ Hint: Notice that each function is half an ellipse.] f( x )= 164 x 2In Problems 65-68, graph each function. Be sure to label all the intercepts. [ Hint: Notice that each function is half an ellipse.] f( x )= 99 x 2In Problems 65-68, graph each function. Be sure to label all the intercepts. [ Hint: Notice that each function is half an ellipse.] f( x )= 6416 x 2In Problems 65-68, graph each function. Be sure to label all the intercepts. [ Hint: Notice that each function is half an ellipse.] f( x )= 44 x 269AYU70AYU71AYU72AYU73AYU74AYU75AYU76AYU77AYU78AYU79AYU80AYU81AYU82AYU83AYU84AYU85AYUThe distance d from P 1 =( 2,5 ) to P 2 =( 4,2 ) is d= _____. (p. 4)To complete the square of x 2 +5x , add ______. (pp. A28-A29)Find the intercepts of the equation y 2 =9+4 x 2 . (pp. 18-19)True or False The equation y 2 =9+ x 2 is symmetric with respect to the x-axis , the y-axis , and the origin. . (pp. 19-21)To graph y= ( x5 ) 3 4 , shift the graph of y= x 3 to the (left/right) _______ unit(s) and then (up/down) _______ unit(s). (pp. 106-114)Find the vertical asymptotes, if any, and the horizontal or oblique asymptotes, if any, of y= x 2 9 x 2 4 . (pp. 224-229)A(n) _______ is the collection of points in a plane the difference of whose distances from two fixed points is a constant.For a hyperbola, the foci lie on a line called the ________.Answer Problems 9-11 using the figure to the right. The equation of the hyperbola is of the form (a) ( xh ) 2 a 2 ( yk ) 2 b 2 =1 (b) ( yk ) 2 a 2 ( xh ) 2 b 2 =1 (c) ( xh ) 2 a 2 + ( yk ) 2 b 2 =1 (d) ( xh ) 2 b 2 + ( yk ) 2 a 2 =1Answer Problems 9-11 using the figure to the right. If the center of the hyperbola is ( 2,1 ) and a=3 , then the coordinates of the vertices are _______ and _________.Answer Problems 9-11 using the figure to the right. If the center of the hyperbola is ( 2,1 ) and c=5 , then the coordinates of the foci are _______ and _________.In a hyperbola, if a=3 and c=5 , then b= ________. (a) 1(b) 2(c) 4(d) 813AYU14AYU15AYUIn Problems 15-18, the graph of a hyperbola is given. Match each graph to its equation. (A) x 2 4 y 2 =1 (B) x 2 y 2 4 =1 (C) y 2 4 x 2 =1 (D) y 2 x 2 4 =117AYU18AYUIn Problems 19-28, find an equation for the hyperbola described. Graph the equation by hand. Center at ( 0,0 ) ; focus at ( 3,0 ) ; vertex at ( 1,0 )In Problems 19-28, find an equation for the hyperbola described. Graph the equation by hand. Center at ( 0,0 ) ; focus at ( 0,5 ) ; vertex at ( 0,3 )In Problems 19-28, find an equation for the hyperbola described. Graph the equation by hand. Center at ( 0,0 ) ; focus at ( 0,6 ) ; vertex at ( 0,4 )In Problems 19-28, find an equation for the hyperbola described. Graph the equation by hand. Center at ( 0,0 ) ; focus at ( 3,0 ) ; vertex at ( 2,0 )In Problems 19-28, find an equation for the hyperbola described. Graph the equation by hand. Foci at ( 5,0 ) ; and ( 5,0 ) ; vertex at ( 3,0 )In Problems 19-28, find an equation for the hyperbola described. Graph the equation by hand. Focus at ( 0,6 ) ; vertices at ( 0,2 ) ; and ( 0,2 )In Problems 19-28, find an equation for the hyperbola described. Graph the equation by hand. vertices at ( 0,6 ) ; and ( 0,6 ) ; asymptote the line y=2xIn Problems 19-28, find an equation for the hyperbola described. Graph the equation by hand. vertices at ( 4,0 ) ; and ( 4,0 ) ; asymptote the line y=2xIn Problems 19-28, find an equation for the hyperbola described. Graph the equation by hand. Foci at ( 4,0 ) ; and ( 4,0 ) ; asymptote the line y=xIn Problems 19-28, find an equation for the hyperbola described. Graph the equation by hand. Foci at ( 0,2 ) ; and ( 0,2 ) ; asymptote the line y=x29AYUIn Problems 29-36, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility. y 2 16 x 2 4 =1In Problems 29-36, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility. 4 x 2 y 2 =1632AYU33AYU34AYU35AYU36AYU37AYUIn Problems 37-40, write an equation for each hyperbola.39AYUIn Problems 37-40, write an equation for each hyperbola.41AYU42AYUIn Problems 41-48, find an equation for the hyperbola described. Graph the equation by hand. Center at (3,4) ; focus at (3,8) ; vertex at (3,2)44AYU45AYU46AYU47AYU48AYU49AYUIn Problems 49-62, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility. ( y+3 ) 2 4 ( x2 ) 2 9 =151AYU52AYU53AYUIn Problems 49-62, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility. ( y3 ) 2 ( x+2 ) 2 =455AYU56AYU57AYU58AYU59AYUIn Problems 49-62, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility. 2 y 2 x 2 +2x+8y+3=061AYUIn Problems 49-62, find the center, transverse axis, vertices, foci, and asymptotes. Graph each equation by hand. Verify your graph using a graphing utility. x 2 3 y 2 +8x6y+4=063AYU64AYU65AYU66AYU67AYUIn Problems 67-74, analyze each equation. ( y+2 ) 2 16 ( x2 ) 2 4 =169AYU70AYUIn Problems 67-74, analyze each equation. 25 x 2 +9 y 2 250x+400=0In Problems 67-74, analyze each equation. x 2 +36 y 2 2x+288y+541=0In Problems 67-74, analyze each equation. x 2 6x8y31=0In Problems 67-74, analyze each equation. 9 x 2 y 2 18x8y88=0Fireworks Display Suppose that two people standing 2 miles apart both see the burst from a fireworks display. After a period of time the first person, standing at point A , hears the burst. One second later the second person, standing at point B , hears the burst. If the person at point B is due west of the person at point A , and if the display is known to occur due north of the person at point A , where did the fireworks display occur?Lightning Strikes Suppose that two people standing 1 mile apart both see a flash of lightning. After a period of time the first person, standing at point A , hears the thunder. Two seconds later the second person, standing at point B , hears the thunder. If the person at point B is due west of the person at point A , and if the lightning strike is known to occur due north of the person standing at point A , where did the lightning strike occur?Nuclear Power Plaut Some nuclear power plants utilize “natural draft� cooling towers in the shape of a hyperboloid, a solid obtained by rotating a hyperbola about its conjugate axis. Suppose that such a cooling tower has a base diameter of 400 feet and the diameter at its narrowest point, 360 feet above the ground, is 200 feet. If the diameter at the top of the tower is 300 feet, how tall is the tower? Source: Bay Area Air Quality Management DistrictAn Explosion Two recording devices are set 2400 feet apart, with the device at point A to the west of the device at point B . At a point between the devices 300 feet from point B , a small amount of explosive is detonated. The recording devices record the time until the sound reaches each. How far directly north of point B should a second explosion be done so that the measured time difference recorded by the devices is the same as that for the first detonation?Rutherford’s Experiment In May 1911, Ernest Rutherford published a paper in Philosophical Magazine. In this article, he described the motion of alpha particles as they are shot at a piece of gold foil 0.00004 cm thick. Before conducting this experiment, Rutherford expected that the alpha particles would shoot through the foil just as a bullet would shoot through snow. Instead, a small fraction of the alpha particles bounced off the foil. This led to the conclusion that the nucleus of an atom is dense, while the remainder of the atom is sparse. Only the density of the nucleus could cause the alpha particles to deviate from their path. The figure shows a diagram from Rutherford’s paper that indicates that the deflected alpha particles follow the path of one branch of a hyperbola. (a) Find an equation of the asymptotes under this scenario. (b) If the vertex of the path of the alpha particles is 10 cm from the center of the hyperbola, find a model that describes the path of the particle.Hyperbolic Mirrors Hyperbolas have interesting reflective properties that make them useful for lenses and mirrors. For example, if a ray of light strikes a convex hyperbolic mirror on a line that would (theoretically) pass through its rear focus, it is reflected through the front focus. This property, and that of the parabola, were used to develop the Cassegrain telescope in 1672. The focus of the parabolic mirror and the rear focus of the hyperbolic mirror are the same point. The rays are collected by the parabolic mirror, then are reflected toward the (common) focus, and thus are reflected by the hyperbolic mirror through the opening to its front focus, where the eyepiece is located. If the equation of the hyperbola is y 2 9 x 2 16 =1 and the focal length (distance from the vertex to the focus) of the parabola is 6, find the equation of the parabola. Source: www.enchantedlearning.comThe eccentricity e of a hyperbola is defined as the number c a , where a is the distance of a vertex from the center and c is the distance of a focus from the center. Because ca , it follows that e1 . Describe the general shape of a hyperbola whose eccentricity is close to 1. What is the shape if e is very large?A hyperbola for which a=b is called an equilateral hyperbola. Find the eccentricity e of an equilateral hyperbola. [Note: The eccentricity of a hyperbola is defined in Problem 81.]Two hyperbolas that have the same set of asymptotes are called conjugate. Show that the hyperbolas x 2 4 y 2 =1 and y 2 x 2 4 =1 are conjugate. Graph each hyperbola on the same set of coordinate axes.Prove that the hyperbola y 2 a 2 x 2 b 2 =1 has the two oblique asymptotes y= a b x and y= a b xShow that the graph of an equation of the form A x 2 +C y 2 +F=0A0,C0,F0 where A and C are opposite in sign, is a hyperbola with center at ( 0,0 ) .Show that the graph of an equation of the form A x 2 +C y 2 +Dx+Ey+F=0,A0,C0 where A and C are opposite in sign, (a) is a hyperbola if D 2 4A + E 2 4C F0 . (b) is two intersecting lines if D 2 4A + E 2 4C F=0 .The sum formula for the sine function is sin( A+B )= . (p.493)2AYU3AYU4AYU5AYU6AYU7AYU8AYU9AYU10AYU11AYU12AYU13AYU14AYU15AYU16AYU17AYU18AYU19AYU20AYU21AYU22AYU23AYU24AYU25AYU26AYU27AYU28AYU29AYU30AYUIn Problems 31-42, rotate the axes so that the new equation contains no xy -term. Analyze and graph the new equation. Refer to Problems 21-30 for Problems 31-40. x 2 +4xy+ y 2 3=0In Problems 31-42, rotate the axes so that the new equation contains no xy -term. Analyze and graph the new equation. Refer to Problems 21-30 for Problems 31-40. x 2 4xy+ y 2 3=0In Problems 31-42, rotate the axes so that the new equation contains no xy -term. Analyze and graph the new equation. Refer to Problems 21-30 for Problems 31-40. 5 x 2 +6xy+5 y 2 8=0In Problems 31-42, rotate the axes so that the new equation contains no xy -term. Analyze and graph the new equation. Refer to Problems 21-30 for Problems 31-40. 3 x 2 10xy+3 y 2 32=035AYUIn Problems 31-42, rotate the axes so that the new equation contains no xy -term. Analyze and graph the new equation. Refer to Problems 21-30 for Problems 31-40. 11 x 2 10 3 xy+ y 2 4=0In Problems 31-42, rotate the axes so that the new equation contains no xy -term. Analyze and graph the new equation. Refer to Problems 21-30 for Problems 31-40. 4 x 2 4xy+ y 2 8 5 x16 5 y=0In Problems 31-42, rotate the axes so that the new equation contains no xy -term. Analyze and graph the new equation. Refer to Problems 21-30 for Problems 31-40. x 2 +4xy+4 y 2 +5 5 y+5=0In Problems 31-42, rotate the axes so that the new equation contains no xy -term. Analyze and graph the new equation. Refer to Problems 21-30 for Problems 31-40. 25 x 2 36xy+40 y 2 12 13 x8 13 y=0In Problems 31-42, rotate the axes so that the new equation contains no xy -term. Analyze and graph the new equation. Refer to Problems 21-30 for Problems 31-40. 34 x 2 24xy+41 y 2 25=0In Problems 31-42, rotate the axes so that the new equation contains no xy -term. Analyze and graph the new equation. Refer to Problems 21-30 for Problems 31-40. 16 x 2 +24xy+9 y 2 130x+90y=0In Problems 31-42, rotate the axes so that the new equation contains no xy -term. Analyze and graph the new equation. Refer to Problems 21-30 for Problems 31-40. 16 x 2 +24xy+9 y 2 60x+80y=0In Problems 43-52, identify the graph of each equation without applying a rotation of axes. x 2 +3xy2 y 2 +3x+2y+5=0In Problems 43-52, identify the graph of each equation without applying a rotation of axes. 2 x 2 3xy+4 y 2 +2x+3y5=0In Problems 43-52, identify the graph of each equation without applying a rotation of axes. x 2 7xy+3 y 2 y10=0In Problems 43-52, identify the graph of each equation without applying a rotation of axes. 2 x 2 3xy+2 y 2 4x2=0In Problems 43-52, identify the graph of each equation without applying a rotation of axes. 9 x 2 +12xy+4 y 2 xy=0In Problems 43-52, identify the graph of each equation without applying a rotation of axes. 10 x 2 +12xy+4 y 2 xy+10=0In Problems 43-52, identify the graph of each equation without applying a rotation of axes. 10 x 2 12xy+4 y 2 xy10=0In Problems 43-52, identify the graph of each equation without applying a rotation of axes. 4 x 2 +12xy+9 y 2 xy=0In Problems 43-52, identify the graph of each equation without applying a rotation of axes. 3 x 2 2xy+ y 2 +4x+2y1=0In Problems 43-52, identify the graph of each equation without applying a rotation of axes. 3 x 2 +2xy+ y 2 +4x2y+10=053AYU54AYU55AYU56AYUUse the rotation formulas ( 5 ) to show that distance is invariant under a rotation of axes. That is, show that the distance from P 1 =( x 1 , y 1 ) to P 2 =( x 2 , y 2 ) in the xy -plane equals the distance from P 1 =( x ' 1 ,y ' 1 ) to P 2 =( x ' 2 ,y ' 2 ) in the x',y' -plane.Show that the graph of the equation x 1/2 + y 1/2 = a 1/2 is part of the graph of a parabola.Formulate a strategy for analyzing and graphing an equation of the form A x 2 +C y 2 +Dx+Ey+F=0Explain how your strategy presented in Problem 59 changes if the equation is of the form A x 2 +Bxy+C y 2 +Dx+Ey+F=01AYUTransform the equation r=6cos from polar coordinates to rectangular coordinates. (pp. 583-584)A _______ is the set of points P in a plane such that the ratio of the distance from a fixed point called the _______ to P to the distance from a fixed line called the _______ to P equals a constant e .The eccentricity e of a parabola is ________, of an ellipse it is _________ and of a hyperbola it is _________.True or False If ( r, ) are polar coordinates, the equation r= 2 2+3sin defines a hyperbola.6AYU7AYU8AYU9AYU10AYU11AYU12AYUIn Problems 13-24, analyze each equation and graph it. r= 1 1+cosIn Problems 13-24, analyze each equation and graph it. r= 3 1sinIn Problems 13-24, analyze each equation and graph it. r= 8 4+3sinIn Problems 13-24, analyze each equation and graph it. r= 10 5+4cos