Skills Practice Skills Practice for Lesson.1 Name Date Conics? Conics as Cross Sections Vocabulary Write the term that best completes each statement. 1. are formed when a plane intersects a double-napped cone. 2. One of two cones separated by a single vertex is called a(n). 3. When the intersection of a plane and a double-napped cone form a point, a line, or intersecting lines, the cross section is called a(n). Problem Set Describe the shape of the cross section created by each intersection described. Draw a figure illustrating the intersection of the two figures and the cross section created. 1. A plane intersects a cube such that the plane is parallel to a face of the cube. The intersection is a square. The square is congruent to a face of the cube. Chapter l Skills Practice 739
2. A plane intersects a cube along the corresponding diagonals of opposite faces of the cube, so that the plane divides the cube into congruent triangular prisms. 3. A plane intersects a sphere such that the intersection is not a point and the plane does not go through the center of the sphere. 70 Chapter l Skills Practice
Name Date. A plane intersects a sphere such that the plane goes through the center of the sphere. 5. A plane intersects a cone such that the plane is parallel to the base of the cone and the intersection is not at the vertex of the cone. Chapter l Skills Practice 71
6. A plane intersects a cone such that the plane is perpendicular to the base of the cone and goes through the vertex of the cone. 7. A plane intersects a cylinder such that the plane is perpendicular to the bases of the cylinder and goes through the centers of both bases. 72 Chapter l Skills Practice
Name Date 8. A plane intersects a cylinder such that the plane is parallel to the bases of the cylinder. 9. A plane intersects a rectangular pyramid such that the plane is parallel to the base of the pyramid and the intersection is not at the vertex of the pyramid. Chapter l Skills Practice 73
10. A plane intersects a rectangular pyramid such that the plane is perpendicular to the base of the pyramid and the plane goes through the vertex of the pyramid.. A plane intersects a cube such that one of the corners of the cube is cut off. 7 Chapter l Skills Practice
Name Date 12. A plane intersects a cylinder such that the plane is neither parallel nor perpendicular to the bases of the cylinder, and the intersection is not a point. Chapter l Skills Practice 75
76 Chapter l Skills Practice
Skills Practice Skills Practice for Lesson.2 Name Date Circles Writing Equations of Circles in General and Standard Form Vocabulary Match each definition to its corresponding term. 1. ( x h) 2 ( y k) 2 r 2, where r represents a. circle the radius of a circle and (h, k) represents the center of the circle 2. the set of all points in a plane that are the b. locus same distance from a given point, called the center 3. Ax 2 By 2 Cx Dy E 0, where A B c. standard form of the equation of a circle. a collection of points that share a property d. general form of the equation of a circle Problem Set Determine whether each equation represents a circle. Explain why or why not. 1. x 2 2y 2 1 0 No, because the coefficients of x and y are not equal. 2. 2x 2 2y 2 10 3. 6y 2 6x 2 2 0 Chapter l Skills Practice 77
. 3x 2 3y 2 9 0 5. 3x 2 2y 2 0 6. y 2 100 x 2 Rewrite the equation for each circle in the form x 2 y 2 r 2. Then calculate the radius. 7. 2x 2 2y 2 18 0 8. y 2 x 2 0 2x 2 2y 2 18 x 2 y 2 9 x 2 y 2 3 2 r 3 9. 0.5x 2 6 0.5y 2 10. 1 3 x2 1 3 y2 1. 7x 2 7y 2 28 0 12. 6x 2 5 6y 2 78 Chapter l Skills Practice
Name Date Write the equation of each circle in general form. 13. x 2 ( y 1) 2 1. ( x 2) 2 y 2 1 x 2 y 2 2y 1 x 2 y 2 2y 3 0 15. ( x 5) 2 ( y 2) 2 10 16. ( x 3) 2 ( y ) 2 37 17. ( x 1 ) 2 ( y 1 2 ) 2 15 18. ( x 1 3 ) 2 ( y 2 3 ) 2 8 Write the equation of each circle in standard form. 19. x 2 y 2 x y 1 0 x 2 y 2 x y 1 x 2 x y 2 y 1 ( x 2) 2 ( y 2) 2 9 20. x 2 y 2 2x 6y 6 0 21. 2x 2 2y 2 8x 8y 2 0 Chapter l Skills Practice 79
22. 3x 2 3y 2 6x 30y 33 0 23. 2x 2 2y 2 5x 9y 16 0 2. x 2 y 2 6x 1y 0 0 Identify the center and radius of each circle. 25. ( x 3) 2 ( y 2) 2 9 26. ( x 1) 2 ( y ) 2 1 The center is (3, 2). The radius is 9 3. 27. x 2 ( y 6) 2 1 28. ( x 5) 2 ( y 5) 2 30 29. ( x 1 7 ) 2 ( y 1 2 ) 2 81 30. ( x 3 ) 2 y 2 8 750 Chapter l Skills Practice
Name Date Sketch a graph of each circle. Label the center. 31. x 2 y 2 25 32. x 2 y 2 9 y 3 2 1 (0, 0) 3 2 1 1 2 3 x 1 2 3 33. ( x 2) 2 y 2 9 3. x 2 ( y 3) 2 25 Chapter l Skills Practice 751
35. ( x ) 2 ( y 2) 2 36. ( x 3) 2 ( y 1) 2 16 Write the equation of each circle in standard form. 37. y 38. y 8 3 6 2 1 3 2 1 1 2 3 x 1 2 3 2 8 6 2 2 6 8 x 2 6 8 x 2 y 2 1 752 Chapter l Skills Practice
Name Date 39. y 0. 8 6 2 y 8 6 2 8 6 2 2 2 6 8 x 8 6 2 2 2 6 8 x 6 6 8 8 1. y 2. y 1 3 12 2 10 1 8 7 6 5 3 2 1 1 x 1 2 3 6 2 6 2 2 6 8 10 x 2 Chapter l Skills Practice 753
75 Chapter l Skills Practice
Skills Practice Skills Practice for Lesson.3 Name Date Your Circle is in my Line, Your Line is in my Circle Intersection of Circles and Lines Problem Set Sketch each system consisting of a circle and a line. Indicate whether the system has zero, one, or two solutions. 1. x 2 y 2 2. ( x 2) 2 y 2 9 y 2 y 8 6 y 1 2 x 7 2 8 6 2 2 2 6 8 x 6 8 The system has one solution. Chapter l Skills Practice 755
3. x 2 ( y 3) 2 16. x 2 y 2 36 y x 1 y 5 2 x 9 5. ( x 2) 2 ( y 3) 2 16 6. ( x ) 2 ( y ) 2 9 y x 3 y 2x 756 Chapter l Skills Practice
Name Date For each system, sketch the circle and the line. Then estimate the solution to the system of equations. Check each solution using the original equations. 7. x 2 y 2 18 8. x 2 y 2 9 y x x y 7 0 y 8 6 2 8 6 2 2 2 6 8 x 6 8 (3, 3) and ( 3, 3) Check: x 2 y 2 18 x 2 y 2 18 (3) 2 (3) 2 18 ( 3) 2 ( 3) 2 18 9 9 18 9 9 18 18 18 18 18 y x y x 3 3 3 3 Chapter l Skills Practice 757
9. ( x 2) 2 y 2 25 x 3y 8 0 758 Chapter l Skills Practice
Name Date 10. x 2 ( y 1) 2 32 x y 1 Chapter l Skills Practice 759
. x 2 x y 2 16 0 y 760 Chapter l Skills Practice
Name Date 12. x 2 y 2 25 y x 1 Chapter l Skills Practice 761
Solve each system of equations algebraically. Check each solution using the original equations. 13. x 2 y 2 13 y x 1 x 2 y 2 13 y x 1 x 2 ( x 1) 2 13 y 2 1 x 2 x 2 2x 1 13 y 3 2x 2 2x 12 0 2( x 2 x 6) 0 y x 1 2( x 3)( x 2) 0 y 3 1 x 2, 3 y 2 Solution: ( 2, 3) and (3, 2) Check: x 2 y 2 13 y x 1 ( 2) 2 ( 3) 2 13 3 2 1 9 13 3 3 13 13 x 2 y 2 13 y x 1 (3) 2 (2) 2 13 2 3 1 9 13 2 2 13 13 762 Chapter l Skills Practice
Name Date 1. x 2 y 2 10 y 3x 10 Chapter l Skills Practice 763
15. x 2 y 2 2y 1 1 y x 6 76 Chapter l Skills Practice
Name Date 16. x 2 ( y 2) 2 25 y 3 Chapter l Skills Practice 765
17. x 2 y 2 100 y x 2 766 Chapter l Skills Practice
Name Date 18. x 2 y 2 18 y x The graph of a circle is represented by two graphs, a top semicircle and a bottom semicircle. Solve each equation for the variable y. Label the top half of the circle y 1 and the bottom half of the circle y 2. 19. x 2 y 2 9 x 2 y 2 9 y 2 9 x 2 y 9 x 2 y 1 9 x 2 y 2 9 x 2 Chapter l Skills Practice 767
20. x 2 y 2 20 21. x 2 ( y 3) 2 81 22. ( x 2) 2 y 2 50 23. ( x ) 2 ( y 10) 2 100 768 Chapter l Skills Practice
Name Date 2. x 2 6x y 2 2y 15 0 Solve each system using a graphing calculator. Provide a sketch of your calculator s viewing screen. 25. x 2 y 2 12 26. ( x 2) 2 y 2 17 y 2x 1 y 1 12 x 2 y 2 12 x 2 y 1 2 x 1 y 3 2 x 1 ( 1.936, 2.872) and (1.136, 3.272) Chapter l Skills Practice 769
27. ( x 3) 2 y 2 21 28. x 2 6x 9 y 2 16 0 y 3x 2 y 2 29. x 2 ( y 2) 2 10 30. x 2 y 2 25 y 1 x 3 y x 5 770 Chapter l Skills Practice
Skills Practice Skills Practice for Lesson. Name Date Going Off on a Tangent (Line) Tangent Lines Problem Set The center of a circle C and a point P on the circle is given. Determine the slope of the line tangent to the circle at point P. 1. C( 1, 2) and P(, 2) The slope of the radius from the center to the point of tangency is 2 2 1 5. The slope of the tangent line is 5 because it is perpendicular to the radius at the point of tangency. 2. C( 3, 3) and P(1, 0) 3. C(6, 1) and P(2, ). C ( 1 2, 3 2 ) and P ( 1, 1 ) 5. C( 2, 2 2 ) and P( 1, 0) Chapter l Skills Practice 771
6. C(0, ) and P(0, 6) The graph of a circle with center C is shown. On the graph draw the tangent line to the circle at point P. Determine the equation of this tangent line. 7. C(0, 0) and P(3, ) y 8 6 2 8. C(0, 0) and P(2, 6) y 8 6 2 8 6 2 2 6 2 6 8 x 8 6 2 2 6 2 6 8 x 8 8 The slope of the radius from the center to the point of tangency is 3. The slope of the tangent line is 3 because it is perpendicular to the radius at the point of tangency. m 3 through (3, ) y mx b 3 (3) b 9 b 25 b y mx b y 3 x 25 772 Chapter l Skills Practice
Name Date 9. C(2, 0) and P(5, 1) 10. C(0, 1) and P(2, 8) y y 8 8 6 6 2 2 8 6 2 2 2 6 8 x 8 6 2 2 2 6 8 x 6 6 8 8 Chapter l Skills Practice 773
. C( 3, 1) and P(2.5, 7) 12. C(2, ) and P ( 1 5, 8 ) y 8 y 8 6 6 2 2 12 10 8 6 2 2 2 x 6 8 6 2 2 2 6 8 x 6 6 8 8 77 Chapter l Skills Practice
Name Date Identify the center and radius of each circle. 13. x 2 6x y 2 y 3 0 1. x 2 2 2 x y 2 2 3 2 0 x 2 6x y 2 y 3 0 ( x 2 6x 9) ( y 2 y ) 16 ( x 3) 2 ( y 2) 2 16 center: (3, 2) and radius: 15. 2x 2 x 2y 2 2y 32 0 16. 16x 2 16x 16y 2 8y 91 0 17. x 2 0.x y 2 0.2y 0. 0 18. 2x 2 12x 2y 2 16y 20 0 Chapter l Skills Practice 775
Determine the equation of the line tangent to the given circle at the given point. 19. Circle: ( x 2) 2 y 2 17 20. Circle: x 2 ( y 3) 2 29 Point of Tangency: (3, ) Point of Tangency: ( 5, 5) Center: (2, 0) Slope of radius to point of tangency: 0 3 2 Slope of tangent line: 1 y mx b 1 (3) b 3 b 19 b y 1 x 19 21. Circle: ( x 0.5) 2 y 2 53 22. Circle: ( x ) 2 ( y 5) 2 0 Point of Tangency: ( 2.5, 7) Point of Tangency: (2, ) 776 Chapter l Skills Practice
Name Date 23. Circle: x 2 6x y 2 9 0 2. Circle: x 2 8x y 2 1y 58.5 0 Point of Tangency: (6, 3) Point of Tangency: ( 3.5,.5) Determine the equation of a circle with the given center that is tangent to the given line. 25. Center: (0, 0) 26. Center: (0, 0) Equation of Tangent Line: y x 8 Slope of radius: 1 Equation of line containing radius: y 0 1( x 0) y x The point (, ) is the intersection of y x 8 and y x: x 8 x Equation of Tangent Line: y 3 8 x 73 8 8 2x x and y Radius: ( 0) 2 ( 0) 2 Equation of circle: x 2 y 2 32 2 Chapter l Skills Practice 777
27. Center: (0, ) 28. Center: ( 3, 0) Equation of Tangent Line: y 5 6 x 37 6 Equation of Tangent Line: y 3x 21 29. Center: (2, 1) 30. Center: (1, 3) Equation of Tangent Line: y 10 Equation of Tangent Line: y 2x 5 778 Chapter l Skills Practice
Skills Practice Skills Practice for Lesson.5 Name Date Circles, Circles, All About Circles Intersections of Two Circles Problem Set Graph each system of circles on the coordinate grid to determine the number of solutions to the system. 1. x 2 y 2 36 2. ( x ) 2 ( y 3) 2 16 ( x 2) 2 y 2 16 ( x 5) 2 y 2 9 y 8 6 2 8 6 2 2 2 6 8 x 6 8 The system has one solution. Chapter l Skills Practice 779
3. x 2 y 2 25. ( x 3) 2 ( y 2) 2 25 ( x 1) 2 ( y 7) 2 36 ( x 10) 2 ( y 2) 2 5. ( x 2) 2 ( y 3) 2 10 6. x 2 6x y 2 6y 13 0 1 2 x2 2x 1 2 y2 3y 9 2 3 x2 x y 2 8y 16 0 780 Chapter l Skills Practice
Name Date Graph each system of circles on the coordinate grid and estimate the solution of the system using the graph. 7. ( x 6) 2 ( y ) 2 0 8. ( x 2) 2 ( y 3) 2 26 ( x 3) 2 ( y 5) 2 58 ( x ) 2 ( y 1) 2 26 y 16 12 8 16 12 8 8 12 16 x 8 12 16 The solutions to the system are (0, 2) and (, 2). 9. ( x 1) 2 ( y 6) 2 153 10. ( x 3) 2 ( y 1) 2 25 ( x 1) 2 ( y 9) 2 18 ( x 5) 2 ( y 3) 2 65 Chapter l Skills Practice 781
. ( x 2) 2 ( y 3) 2 16 12. ( x ) 2 ( y 3) 2 25 ( x 5) 2 ( y ) 2 9 ( x 3) 2 ( y 3) 2 13. ( x 5) 2 ( y 5) 2 26 1. x 2 8x y 2 2y 3 0 ( x 1) 2 ( y 1) 2 26 x 2 6x y 2 2y 31 0 782 Chapter l Skills Practice
Name Date Solve each system of circles algebraically using the elimination process. Include a sketch of each system. If the system has a solution, label the points. 15. x 2 y 2 1 ( x 1) 2 y 2 50 y 8 6 (, 5) 2 8 6 2 2 2 6 8 x 6 (, 5) 8 x 2 y 2 1 ( x 1) 2 y 2 50 x 2 y 2 1 x 2 2x 1 y 2 50 x 2 y 2 1 x 2 2x y 2 9 x 2 y 2 1 x 2 2x y 2 9 2 x 8 x x 2 y 2 1 2 y 2 1 16 y 2 1 y 2 25 y 5 The solutions to the system are (, 5) and (, 5). Chapter l Skills Practice 783
16. x 2 y 2 25 x 2 y 2 9 78 Chapter l Skills Practice
Name Date 17. ( x ) 2 y 2 16 ( x 2) 2 y 2 Chapter l Skills Practice 785
18. ( x 2) 2 ( y ) 2 16 3x 2 12x 3y 2 2y 12 0 786 Chapter l Skills Practice
Name Date 19. ( x 5) 2 ( y 5) 2 65 ( x 3) 2 ( y 7) 2 65 Chapter l Skills Practice 787
20. x 2 y 2 3x 15 x 2 y 2 96 788 Chapter l Skills Practice
Name Date 21. ( x 2) 2 ( y 3) 2 16 ( x 1) 2 ( y 2) 2 9 Chapter l Skills Practice 789
22. ( x 3) 2 ( y 3) 2 32 ( x ) 2 ( y ) 2 18 790 Chapter l Skills Practice
Name Date Solve each system of circles using a graphing calculator. Include a sketch of your calculator s viewing screen. 23. x 2 y 2 25 2. x 2 y 2 16 x 2 ( y 6) 2 ( x 5) 2 y 2 y 1 25 x 2 y 2 25 x 2 y 3 6 x 2 y 6 x 2 The solutions to the system are the points ( 1.561,.75) and (1.561,.75). 25. ( x 3) 2 ( y 3) 2 16 26. ( x 2) 2 ( y 3) 2 16 ( x 2) 2 ( y 3) 2 25 ( x 2) 2 ( y 3) 2 25 Chapter l Skills Practice 791
27. ( x 2) 2 ( y 3) 2 25 28. ( x ) 2 ( y 3) 2 9 ( x 2) 2 ( y 5) 2 9 ( x 5) 2 ( y ) 2 16 792 Chapter l Skills Practice
Skills Practice Skills Practice for Lesson.6 Name Date Get Into Gear Circles and Problem Solving Problem Set Use the given information to answer each question. 1. Maya wants to plant trees in a circular pattern 30 feet from her house. To get an idea of how this would look, she placed the location of her house at the origin of a coordinate system and then drew a circle about the location of her house with a radius of 30 feet. Draw a diagram to represent Maya s diagram, and determine the equation of the circle on which the trees will be planted. Write your equation in general form. x 2 y 2 900 0 y 0 30 20 10 0 30 20 10 10 10 20 30 0 x 20 30 0 2. In pre-school, Mrs. Ono taught her students how to play Circle That Student. One student stands in the middle of a circle of students while the other students hold hands and move on the path of the circle while music plays. When the music stops the students all fall down. If the student standing in the middle of the other students is located at the origin and the other students are 3.5 meters away, what is the equation of the circular path the other students are moving on? Write your equation in standard form. Chapter l Skills Practice 793
3. Farmer Mills is planning on putting a circular fishing pond on his property. He lays out his plan on a coordinate system with the farmhouse at the origin and the center of the pond located at the point (700, 900). This means that the center of the pond will be located 700 yards due east and then 900 yards due north of his farmhouse. If the pond is to have a radius of 150 yards, what is the equation of the circle representing the pond? Write your equation in general form.. A landscaper is planning a circular flower bed. She first makes a sketch of the flower bed on a coordinate system with the center of the flower bed at the origin with a radius of 5.2 feet. Not liking what she sees, she moves the location of the flower bed so that the new center is feet to the left and 7.1 feet down. Write the equation of the circle that represents the new flower bed in standard form. 5. In math class, Kendra was asked to graph the circle represented by the equation x 2 1x y 2 8y 0 0. Her teacher instructed her to write the equation in standard form, identify the center and radius, and graph it. Kendra knows that she must show all her work. 79 Chapter l Skills Practice
Name Date 6. In drafting class, Ms. Stewart asked each of her students to draw a circle centered at the origin with a radius of 5 centimeters on a coordinate system. She then instructed them to draw the circle represented by the equation ( x ) 2 ( y 6) 2 25 on the same coordinate system. She then called on Eli to explain how the two circles are related. If Eli answers correctly, what might he have said? Determine the solution set for each of the given systems of equations. 7. The park district is holding a concert on one of their softball fields. To keep the crowd back, they erected a fence on the field by driving posts into the ground and then nailing boards to the posts. Prior to erecting the fence, they drew a schematic diagram of the project. As you can see, two of the posts are located on the perimeter of the field. What are the coordinates for the location of these posts, if the equation for the perimeter of the field can by represented by the equation x 2 y 2 13,000 and the fence posts are located on the line represented by the equation y 3 x? x 2 y 2 13,000 y 3 x x 2 ( 3 x ) 2 13,000 y 3 (0) x 2 9 121 x2 13,000 y 30 121x 2 9x 2 1,573,000 130x 2 1,573,000 y 3 ( 0) x 2 12,100 y 30 x 0 The coordinates where the posts are located are (0, 30) and ( 0, 30). Chapter l Skills Practice 795
8. A mosaic, as shown in the diagram, consists of a circle inscribed in a square. The equation of the circle is given by x 2 y 2.5. What are the coordinates of the point of tangency of the circle and square in the first quadrant? y (0, 3) ( 3, 0) (3, 0) x (0, 3) 796 Chapter l Skills Practice
Name Date 9. The ceiling of a modern building is made up of a circular window with a beam passing through the circle supporting the glass. The equation for the circle of glass is ( x 5) 2 ( y ) 2 65, while the beam lies on a line with equation y 3 2 x 3 2. What are the coordinates of the points where the beam intersects the perimeter of the glass? Chapter l Skills Practice 797
10. An indoor field house is circular in shape and can be modeled by the equation x 2 y 2 80. A young boy, standing in the field house at what would be the point (, 2) if the equation of the circle were graphed and his position located, rolls a ball through the center of the circle. If the ball hits the wall of the field house, at what point on the circle does the ball intersect? 798 Chapter l Skills Practice
Name Date. Center City is unique in that it has a fountain at its center and a circular drive about the fountain. As seen in the schematic diagram, there are four roads entering the circular drive; one each from the northeast, northwest, southeast, and southwest. Suppose the circular drive can be modeled by the equation ( x ) 2 y 2 6 and the road entering the city from the southeast went directly through the fountain and joined up with the road entering the city from the northwest. Determine an equation of the line that best represents this proposed road and determine the points of intersection of this line with the equation representing the circular drive. NW NE Fountain SW SE Chapter l Skills Practice 799
12. Aiden ties a tennis ball to a string and begins to swing the ball over his head in a circular pattern modeled by the equation ( x 2) 2 ( y 3) 2 36. As Aiden increases the speed of the tennis ball, the string breaks and the ball follows a linear path that can be modeled by the equation y 1 12 x 17. If the string broke at a 6 point where the x and y values are integers, at what point on the circular path of the ball did the string break? 800 Chapter l Skills Practice
Name Date Use the given information to determine each equation. 13. Jordan is in Algebra class. Mr. Snow, her teacher, asks Jordan to determine the equation of a line tangent to a circle with center (5, 3) at the point (2, ). Determine the slope of the radius of the circle at the point of tangency. ( 3) m 2 5 1 3 The slope of the tangent line is the negative reciprocal of 1 or 3. 3 The equation of the tangent line is found as follows. y ( ) 3( x 2) y 3x 6 y 3x 2 Chapter l Skills Practice 801
1. Elvia wants to work in a circus. Her talent lies in her exceptional ability to balance a board on a ball while standing on the board. If Elvia is doing this correctly the board should be parallel to the ground, but as you can see it is not. In fact the path of the board is modeled by the linear equation y 2 x. What is the equation of the circle that represents the outline 3 of the ball? C(7, 2) 802 Chapter l Skills Practice
Name Date 15. Buster s Café has a unique sign out front. The circular body of the sign can be modeled by the equation ( x 1) 2 y 2 29 and the arrow is tangent to the circle at the point ( 3, 5). What is the equation of the line that serves as a model for the arrow? Buster s Café Chapter l Skills Practice 803
16. A bowling ball, with center (3, 5), is rolling along a lane which can be modeled by the line y 2. What is the equation of the circle that represents the perimeter of the ball? 80 Chapter l Skills Practice
Name Date 17. Manuel participates in track and field in the hammer throw event. Essentially this is a chain with an iron ball attached to it. Manuel swings the hammer in a circular pattern about his head and releases it. The path of the hammer is linear and perpendicular to the radius of the circular pattern when it leaves his hands. Suppose the circular path of the hammer is modeled by the equation ( x ) 2 ( y 2) 2 0 and the ball leaves this path at the point (6, ). What is the slope of the path of the hammer as it is released? Chapter l Skills Practice 805
18. Harmon is having a difficult time determining the equation of a line tangent to a circle when the point of tangency and the center of the circle are known. Explain to Harmon how to determine the equation of a line tangent to a circle with center ( 2, 0) at the point (5, ). 806 Chapter l Skills Practice
Name Date Use the given information to answer each question. 19. Spencer enjoys playing the game Smash-Up Derby. The game is played on two circular intersecting race tracks. The object is to place a race car on each track and count the number of times each car can complete a circle before they collide. If one of the circular tracks can be modeled by the equation ( x 6) 2 ( y 5) 2 15 and the other by ( x 10) 2 ( y 3) 2 65, at what points of intersection might the cars collide? ( x 6) 2 ( y 5) 2 15 ( x 10) 2 ( y 3) 2 65 x 2 12x 36 y 2 10y 25 15 x 2 20x 100 y 2 6y 9 65 x 2 12 x 36 y 2 10y 25 15 x 2 20x 100 y 2 6y 9 65 32x 16y 128 0 2x y 8 0 y 2x 8 ( x 6) 2 ( y 5) 2 15 ( x 6) 2 [( 2x 8) 5] 2 15 ( x 6) 2 ( 2 x 13) 2 15 x 2 12x 36 x 2 52x 169 15 5x 2 0x 60 0 x 2 8x 12 0 ( x 6)( x 2) 0 x 6 and 2 x 6: x 2: y 2 x 8 y 2 x 8 y 2(6) 8 y 2(2) 8 y y The cars might collide at points (6, ) or (2, ). Chapter l Skills Practice 807
20. Carie graphed the equations x 2 ( y 2) 2 25 and y 3 x 33 on her graphing calculator and showed the results, seen in the figure, to her friend Linda. Linda looked at the graph and told Carie that the solution to the system consisted of two points. Is Linda correct? Explain. 808 Chapter l Skills Practice
Name Date 21. Brodie s little brother and sister are riding their tricycles in circles. The path that Brodie s brother is riding on can be modeled by the equation ( x ) 2 y 2, and the path his sister is riding on can be modeled by the equation ( x 10) 2 y 2. Will his sister and brother ever run into one another? Why or why not? 22. An old gold mining map indicates that gold is buried on the line that passes through the points of intersection of the circles with equations ( x 6) 2 ( y 6) 2 82 and ( x ) 2 ( y 2) 2 82. Determine the equation of the line on which the gold is supposed to be buried. Chapter l Skills Practice 809
23. A hawk and a dove are both flying a circular pattern. The hawk s path can be modeled by the equation ( x ) 2 ( y 2) 2 10, while the dove s path can be modeled by the equation x 2 8x y 2 y 12 0. Will the hawk and dove ever cross paths? Explain. 2. Two marching teams are performing their marching routines. Concurrently one team is marching in a circle that can be modeled by the equation ( x 8) 2 y 2 6 and the other team is marching in a circle that can be modeled by the equation ( x ) 2 y 2 16. Explain why there might be a problem, since the two teams are marching at the same time. 810 Chapter l Skills Practice