1-5 Circles in the Coordinate Plane Common Core State Standards G-GPE.A.1 Derive the equation of a circle given center and radius using the Pthagorean Theorem... MP 1, MP 3, MP, MP 7 bjectives To write the equation of a circle To find the center and radius of a circle Do ou need to check the distance to ever part of the course? MATHEMATICAL PRACTICES The owners of an outdoor adventure course want a wa to communicate to all points on the course. The are considering purchasing walkietalkies with a range of 1 mi. A model of the course is at the right. Each grid unit represents 1 mi. The 8 base station is at (, ). Do ou think the owners should bu the walkietalkies? Wh? Mud pit 8 Ropes course Rock climbing Base station Tire course 6 8 Lesson L VocabularV standard form of an equation of a circle In the Solve It, all of the obstacles lie within or on a circle with the base station as the center. The information from the diagram is enough to write an equation for the circle. Essential Understanding The information in the equation of a circle allows ou to graph the circle. Also, ou can write the equation of a circle if ou know its center and radius. Theorem 1-16 Equation of a Circle An equation of a circle with center (h, k) and radius r is ( - h) + ( - k) = r. r (, ) (h, k) 798 Chapter 1 Circles
Here s Wh It Works You can use the Distance Formula to find an equation of a circle with center (h, k) and radius r, which proves Theorem 1-16. Let (, ) be an point on the circle. Then the radius r is the distance from (h, k) to (, ). (, ) r (h, k) d = ( - 1 ) + ( - 1 ) Distance Formula r = ( - h) + ( - k) Substitute (, ) for (, ) and (h, k) for ( 1, 1 ). r = ( - h) + ( - k) Square both sides. The equation ( - h) + ( - k) = r is the standard form of an equation of a circle. You ma also call it the standard equation of a circle. What do ou need to know to write the equation of a circle? You need to know the values of h, k, and r; h is the -coordinate of the center, k is the -coordinate of the center, and r is the radius. Problem 1 Writing the Equation of a Circle What is the standard equation of the circle with center (5, ) and radius 7? ( - h) + ( - k) = r Use the standard form of an equation of a circle. ( - 5) + [ - (-)] = 7 Substitute (5, -) for (h, k) and 7 for r. Got It? ( - 5) + ( + ) = 9 Simplif. 1. What is the standard equation of each circle? a. center (3, 5); radius 6 b. center (-, - 1); radius 1 Problem Using the Center and a Point on a Circle How is this problem different from Problem 1? In this problem, ou don t know r. So the first step is to find r. What is the standard equation of the circle with center (1, 3) that passes through the point (, )? Step 1 Use the Distance Formula to find the radius. r = ( - 1 ) + ( - 1 ) Use the Distance Formula. = (1 - ) + (-3 - ) Substitute (1, -3) for (, ) and (, ) for ( 1, 1 ). = (-1) + (-5) Simplif. 6 (, ) (1, ) = 16 Step Use the radius and the center to write an equation. ( - h) + ( - k) = r Use the standard form of an equation of a circle. ( - 1) + [ - (-3)] = (16) Substitute (1, -3) for (h, k) and 16 for r. ( - 1) + ( + 3) = 6 Simplif. Got It?. What is the standard equation of the circle with center (, 3) that passes through the point (-1, 1)? Lesson 1-5 Circles in the Coordinate Plane 799
If ou know the standard equation of a circle, ou can describe the circle b naming its center and radius. Then ou can use this information to graph the circle. Problem 3 Graphing a Circle Given Its Equation Communications When ou make a call on a cell phone, a tower receives and transmits the call. A wa to monitor the range of a cell tower sstem is to use equations of circles. Suppose the equation ( - 7) + ( + ) = 6 represents the position and the transmission range of a cell tower. What is the graph that shows the position and range of the tower? STEM The equation representing the cell tower s position and range To draw a graph Determine the values of (h, k) and r in the equation. Then draw a graph. ( 7) ( ) 6 ( 7) [ ( )] 8 c c h r The center is (7, -) and the radius is 8. Use the standard equation of a circle. Rewrite to find h, k, and r. To graph the circle, place the compass point at the center (7, -) and draw a circle with radius 8. Got It? 3. a. In Problem 3, what does the center of the circle represent? What does the radius represent? b. What is the center and radius of the circle with equation ( - ) + ( - 3) = 100? Graph the circle. 8 8 1 1 Lesson Check Do ou know HW? What is the standard equation of each circle? 1. center (0, 0); r =. center (1, -1); r = 15 What is the center and radius of each circle? 3. ( - 8) + = 9. ( + ) + ( - ) = 7 Do ou UNDERSTAND? MATHEMATICAL PRACTICES 5. What is the least amount of information that ou need to graph a circle? To write the equation of a circle? 6. Suppose ou know the center of a circle and a point on the circle. How do ou determine the equation of the circle? 7. Error Analsis A student sas that the center of a circle with equation ( - ) + ( + 3) = 16 is (-, 3). What is the student s error? 800 Chapter 1 Circles
Practice and Problem-Solving Eercises MATHEMATICAL PRACTICES A Practice Write the standard equation of each circle. See Problem 1. 8. center (, -8); r = 9 9. center (0, 3); r = 7 10. center (0., 1.1); r = 0. 11. center (5, -1); r = 1 1. center (-6, 3); r = 8 13. center (-9, -); r = 15 1. center (0, 0); r = 15. center (-, 0); r = 3 16. center (-1, -1); r = 1 Write a standard equation for each circle in the diagram at the right. 17. }P 18. }Q Write the standard equation of the circle with the given center that passes through the given point. 19. center (-, 6); point (-, 10) 0. center (1, ); point (0, 6) 1. center (7, -); point (1, -6). center (-10, -5); point (-5, 5) 3. center (6, 5); point (0, 0). center (- 1, - ); point (-, 0) Find the center and radius of each circle. Then graph the circle. 5. ( + 7) + ( - 5) = 16 6. ( - 3) + ( + 8) = 100 7. ( + ) + ( - 1) = 5 8. + = 36 Public Safet Each equation models the position and range of a tornado alert siren. Describe the position and range of each. 9. ( - 5) + ( - 7) = 81 30. ( + ) + ( - 9) = 1 P See Problem. Q See Problem 3. B Appl Write the standard equation of each circle. 31. 3. 33. 3 3 3. 3 35. 36. Lesson 1-5 Circles in the Coordinate Plane 801
C Challenge Write an equation of a circle with diameter AB. 37. A(0, 0), B(8, 6) 38. A(3, 0), B(7, 6) 39. A(1, 1), B(5, 5) 0. Reasoning Describe the graph of + = r when r = 0. Determine whether each equation is the equation of a circle. Justif our answer. 1. ( - 1) + ( + ) = 9. + = 9 3. + ( - 3) = 9. Think About a Plan Find the circumference and area of the circle whose equation is ( - 9) + ( - 3) = 6. Leave our answers in terms of p. What essential information do ou need? What formulas will ou use? 5. Write an equation of a circle with area 36p and center (, 7). 6. What are the - and -intercepts of the line tangent to the circle ( - ) + ( - ) = 5 at the point (5, 6)? 7. For ( - h) + ( - k) = r, show that = r - ( - h) + k or = - r - ( - h) + k. Sketch the graphs of each equation. Find all points of intersection of each pair of graphs. 8. + = 13 9. + = 17 50. + = 8 = - + 5 = - 1 = 51. + = 0 5. ( + 1) + ( - 1) = 18 53. ( - ) + ( - ) = 10 = - 1 + 5 = + 8 = - 1 3 + 6 5. You can use completing the square and factoring to find the center and radius of a circle. a. What number c do ou need to add to each side of the equation + 6 + - = - so that + 6 + c can be factored into a perfect square binomial? b. What number d do ou need to add to each side of the equation + 6 + - = - so that - + d can be factored into a perfect square binomial? c. Rewrite + 6 + - = - using our results from parts (a) and (b). d. What are the center and radius of + 6 + - = -? e. What are the center and radius of + + - 0 + 100 = 0? 55. The concentric circles ( - 3) + ( - 5) = 6 and ( - 3) + ( - 5) = 5 form a ring. The lines = 3 + 3 and = 5 intersect the ring, making four sections. Find the area of each section. Round our answers to the nearest tenth of a square unit. 80 Chapter 1 Circles
56. Geometr in 3 Dimensions The equation of a sphere is similar to the equation of a circle. The equation of a sphere with center (h, j, k) and radius r is ( - h) + ( - j) + (z - k) = r. M(-1, 3, ) is the center of a sphere passing through T(0, 5, 1). What is the radius of the sphere? What is the equation of the sphere? 57. Nautical Distance A close estimate of the radius of Earth s equator is 3960 mi. a. Write the equation of the equator with the center of Earth as the origin. b. Find the length of a 1 arc on the equator to the nearest tenth of a mile. c. Histor Columbus planned his trip to the East b going west. He thought each 1 arc was 5 mi long. He estimated that the trip would take 1 das. Use our answer to part (b) to find a better estimate. z M T Standardized Test Prep SAT/ACT 58. What is an equation of a circle with radius 16 and center (, -5)? ( - ) + ( + 5) = 16 ( + ) + ( - 5) = 56 ( + ) + ( - 5) = ( - ) + ( + 5) = 56 59. What can ou NT conclude from the diagram at the right? c = d a = b c + e = b e = d c b a e d Short Response 60. Are the following statements equivalent? In a circle, if two central angles are congruent, then the have congruent arcs. In a circle, if two arcs are congruent, then the have congruent central angles. Mied Review Find the value of each variable. See Lesson 1-. 61. 30 6. 80 70 a 1 18 Get Read! To prepare for Lesson 1-6, do Eercises 63 65. See Lessons 1- and 1-5. Sketch each of the following. 63. the perpendicular bisector of BC 6. line k parallel to line m and perpendicular to line w, all in plane N 65. EFG bisected b FH > Lesson 1-5 Circles in the Coordinate Plane 803