TEKS 9.3 a.5, A.5.B Graph and Write Equations of Circles Before You graphed and wrote equations of parabolas. Now You will graph and write equations of circles. Wh? So ou can model transmission ranges, as in E. 6. Ke Vocabular circle center radius A circle is the set of all points (, ) in a plane that are equidistant from a fied point, called the center of the circle. The distance r between the center and an point (, ) on the circle is the radius. For a circle with center at the origin and radius r, the distance between an point (, ) on the circle and the center (0, 0) is r, so the following is true: Ï }} ( 0) ( 0) 5 r Distance formula ( 0) ( 0) 5 r Square each side. r (, ) 5 r 5 r Simplif. KEY CONCEPT For Your Notebook Standard Equation of a Circle with Center at the Origin The standard form of the equation of a circle with center at (0, 0) and radius r is as follows: 5 r E XAMPLE Graph an equation of a circle Graph 5 36. Identif the radius of the circle. STEP STEP STEP 3 Rewrite the equation 5 36 in standard form as 5 36. Identif the center and radius. From the equation, the graph is a circle centered at the origin with radius r 5 Ï } 36 5 6. Draw the circle. First plot several convenient points that are 6 units from the origin, such as (0, 6), (6, 0), (0, 6), and (6, 0). Then draw the circle that passes through the points. 5 36 (0, 6) 4 (6, 0) (6, 0) (0, 6) at classzone.com 66 Chapter 9 Quadratic Relations and Conic Sections
E XAMPLE Write an equation of a circle The point (, 5) lies on a circle whose center is the origin. Write the standard form of the equation of the circle. Because the point (, 5) lies on the circle, the circle s radius r must be the distance between the center (0, 0) and (, 5). Use the distance formula. r 5 Ï }} ( 0) (5 0) 5 Ï } 4 5 5 Ï } 9 The radius is Ï } 9. Use the standard form with r 5 Ï } 9 to write an equation of the circle. 5 r Standard form 5 (Ï } 9 ) Substitute Ï } 9 for r. 5 9 Simplif. E XAMPLE 3 TAKS PRACTICE: Multiple Choice What is an equation of the line tangent to the circle 5 0 at (, 3)? ELIMINATE CHOICES In Eample 3, ou can eliminate choice D because a quick sketch of the circle shows that the slope of the tangent line at (, 3) must be positive. A 5 3 4 B 5 3 8 } 3 C 5 } 3 0 } 3 D 5 } 3 8 } 3 From geometr, a line tangent to a circle is perpendicular to the radius at the point of tangenc. The radius with endpoint (, 3) has slope m 5} 3 0 53, so the slope of the tangent line 0 at (, 3) is the negative reciprocal of 3, or }. 3 An equation of the tangent line is as follows: 3 5 } 3 ( ()) Point-slope form 4 (, 3) 5 0 3 5 } 3 } 3 Distributive propert 5 } 3 0 } 3 Solve for. c The correct answer is C. A B C D GUIDED PRACTICE for Eamples,, and 3 Graph the equation. Identif the radius of the circle.. 5 9. 5 49 3. 8 5 4. Write the standard form of the equation of the circle that passes through (5, ) and whose center is the origin. 5. Write an equation of the line tangent to the circle 5 37 at (6, ). 9.3 Graph and Write Equations of Circles 67
CIRCLES AND INEQUALITIES The regions inside and outside the circle 5 r can be described b inequalities, with < r representing the region inside the circle and > r representing the region outside the circle. > r < r E XAMPLE 4 Write a circular model CELL PHONES A cellular phone tower services a 0 mile radius. You get a flat tire 4 miles east and 9 miles north of the tower. Are ou in the tower s range? STEP STEP Write an inequalit for the region covered b the tower. From the diagram, this region is all points that satisf the following inequalit: < 0 Substitute the coordinates (4, 9) into the inequalit from Step. < 0 Inequalit from Step < 0 (4, 9) 4 9 <? 0 Substitute for and. c So, ou are in the tower s range. 97 < 00 The inequalit is true. In the diagram above, the origin represents the tower and the positive -ais represents north. E XAMPLE 5 Appl a circular model CELL PHONES In Eample 4, suppose that ou fi our tire and then drive south. For how man more miles will ou be in range of the tower? When ou leave the tower s range, ou will be at a point on the circle 5 0 whose -coordinate is 4 and whose -coordinate is negative. Find the point (4, ) where < 0 on the circle 5 0. 5 0 Equation of the circle (4, 9) 4 5 0 Substitute 4 for. 56Ï } 84 Solve for. ø 69. Use a calculator. c Because < 0, ø 9.. You will be in the tower s range from (4, 9) to (4, 9.), a distance of 9 (9.) 5 8. miles. (4, ) GUIDED PRACTICE for Eamples 4 and 5 6. WHAT IF? In Eamples 4 and 5, suppose ou drive west after fiing our tire. For how man more miles will ou be in range of the tower? 68 Chapter 9 Quadratic Relations and Conic Sections
9.3 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 7, 39, and 65 5 TAKS PRACTICE AND REASONING Es., 43, 59, 64, 66, 70, and 7 5 MULTIPLE REPRESENTATIONS E. 68 EXAMPLE on p. 66 for Es. 3. VOCABULARY The radius of a circle is the distance from an point on the circle to a fied point called the circle s?.. WRITING How are the slope of a line tangent to a circle and the slope of the radius at the point of tangenc related? MATCHING GRAPHS Match the equation with its graph. 3. 5 9 4. 5 36 5. 5 4 6. 5 6 7. 5 6 8. 5 3 A. B. 3 C. 3 D. E. F. GRAPHING Graph the equation. Identif the radius of the circle. 9. 5 0. 5 8. 5 5. 5 3. 5 7 4. 5 40 5. 5 5 6. 5 9 7. 5 5 5 60 8. 7 7 5 9. 4 4 5 8 0. 8 8 5 9. TAKS REASONING What is the radius of the circle 3 3 5 54? A 3Ï } B 3Ï } 6 C 8 D 54 EXAMPLE on p. 67 for Es. 43 WRITING EQUATIONS Write the standard form of the equation of the circle with the given radius and whose center is the origin.. 3. 8 4. 5. 6 6. Ï } 7. Ï } 5 8. 5Ï } 9. 4Ï } 6 30. ERROR ANALYSIS Describe and correct the error in writing an equation of the circle with the given center and radius. Center: (0, 0); Radius: Equation: 5 9.3 Graph and Write Equations of Circles 69
WRITING EQUATIONS Write the standard form of the equation of the circle that passes through the given point and whose center is the origin. 3. (6, 0) 3. (0, 5) 33. (4, 3) 34. (, 4) 35. (6, 8) 36. (9, ) 37. (4, 0) 38. (8, 5) 39. (8, 4) 40. (5, ) 4. (, ) 4. (9, 40) 43. TAKS REASONING What is the equation in standard form of the circle that passes through the point (4, 6) and whose center is the origin? A 5 5 B 5 0 C 5 5 D 5 Ï } 3 GRAPHING In Eercises 44 5, equations of both circles and parabolas are given. Graph the equation. 44. 5 49 45. 4 5 0 46. 7 7 5 63 47. 5 48. 6 5 0 49. 3 5 50. 5 9 5. 5 6 5. 6 6 5 0 EXAMPLE 3 on p. 67 for Es. 53 58 TANGENT LINES Write an equation of the line tangent to the given circle at the given point. 53. 5 7; (, 4) 54. 5 3; (, 3) 55. 5 34; (5, 3) 56. 5 40; (6, ) 57. 5 06; (5, 9) 58. 5 50; (5, 5) 59. TAKS REASONING Write equations in standard form for three circles centered at the origin so that each circle passes between (3, 5) and (6, ). 60. REASONING Use the diagram to show that an angle inscribed in a semicircle is a right angle. (Hint: Show that the segments meeting at (, ) have slopes that are negative reciprocals.) (, ) (r, 0) (r, 0) 6. CHALLENGE Suppose two congruent circles intersect so that each passes through the other s center, as shown. Write an equation that gives the length l of the chord formed b joining the intersection points in terms of the radius r of each circle. r l PROBLEM SOLVING EXAMPLE 4 on p. 68 for Es. 6 64 6. CELL PHONES A cellular phone tower services a 5 mile radius. On a hiking trip, ou are 9 miles east and miles north of the cell tower. Are ou in the region served b the tower? 63. BATS During the warmer months, more than million Meican free-tailed bats live under the Congress Avenue Bridge in Austin, Teas. The bats have an estimated feeding range of 50 miles. Is a location 40 miles north and 5 miles west of the bridge located within this range? 630 5 WORKED-OUT SOLUTIONS on p. WS 5 TAKS PRACTICE AND REASONING 5 MULTIPLE REPRESENTATIONS
64. TAKS REASONING An appliance store claims to provide free deliver up to 00 miles from the store. The following points represent the locations of houses, with the origin representing the store. (All coordinates are in miles.) Which house is located outside the free deliver area? A (95, 30) B (90, 35) C (80, 55) D (75, 70) EXAMPLE 5 on p. 68 for Es. 65 67 65. MULTI-STEP PROBLEM Class B airspace sometimes consists of a stack of clindrical laers as shown. Seen from above, the airspace forms circles whose origin is the control tower. A plane in straight and level flight flies through the top laer along the line 54, as shown. a. For how man miles will the plane be in the top-most laer of Class B airspace? b. For how man miles will the plane be above the middle laer of Class B airspace? c. For how man miles will the plane be above the lowest laer of Class B airspace? 66. TAKS REASONING A circular utilit tunnel 8 feet in diameter has a 6-foot-wide walkwa across its bottom. Could a worker who is 6 feet inches tall walk down the center of the walkwa without ducking? Eplain. (Hint: Write an equation of the tunnel s cross section. Find the -coordinate of an endpoint of the walkwa and substitute to find the -coordinate.) 67. GROUNDSKEEPING A row of sprinklers is to be installed parallel to and 4.5 feet awa from the back edge of a flower bed. Each sprinkler waters a region with a 6 foot radius. How far apart should the sprinklers be placed to water the entire flower bed with the least possible overlap in coverage, as shown? 6 ft 4.5 ft?? 68. MULTIPLE REPRESENTATIONS The Modified Mercalli Intensit Scale rates an earthquake s shaking strength. In general, the rating decreases as distance from the earthquake s epicenter increases. Suppose an earthquake has a Mercalli rating of 6.0 at its epicenter, a 5.7 rating 5 miles awa from the epicenter, a 5.4 rating 5 miles awa, and a 5. rating 35 miles awa. a. Drawing Graphs Represent the situation described above using circles in a coordinate plane. b. Writing Inequalities For each circle from part (a), write an inequalit describing the coordinates of locations with a Mercalli rating at least as great as the Mercalli rating represented b the circle. c. Making a Prediction What can ou predict about the Mercalli rating miles west and 6 miles south of the epicenter? Eplain. 9.3 Graph and Write Equations of Circles 63
69. CHALLENGE Two radio transmitters, one with a 40 mile range and one with a 60 mile range, stand 80 miles apart. You are driving 60 miles per hour on a highwa parallel to the line segment connecting the two towers. How long will ou be within range of both transmitters simultaneousl? 40 mi 80 mi highwa 60 mi 0 mi MIXED REVIEW FOR TAKS TAKS PRACTICE at classzone.com REVIEW Lesson 3.; TAKS Workbook 70. TAKS PRACTICE How man solutions does the sstem of equations below have? TAKS Obj. 4 3 57 8 5 8 A None B One C Two D Infinitel man REVIEW Lesson 9.; TAKS Workbook 7. TAKS PRACTICE What is the approimate perimeter of quadrilateral PQRS? TAKS Obj. 7 F.7 units G 6.4 units H 0.8 units J.5 units P 43 3 4 S 4 3 ΠR QUIZ for Lessons 9. 9.3 Find the distance between the two points. Then find the midpoint of the line segment joining the two points. (p. 64). (4, 3), (8, 7). (, 5), (4, 9) 3. (5, ), (4, 8) 4. (, ), (7, ) 5. (6, 5), (, 8) 6. (3, ), (6, 5) Write the standard form of the equation of the parabola with the given focus and verte at (0, 0). (p. 60) 7. (0, 3) 8. (, 0) 9. (6, 0) 0. (0, 4). (0, 5). (, 0) Graph the equation. Identif the radius of the circle. (p. 66) 3. 5 4 4. 5 64 5. 5 0 6. 5 75 7. 3 3 5 48 8. 6 6 5 08 9. ASTRONOMY If the plane in which Jupiter orbits the sun is a coordinate plane with its origin at the sun and coordinates in millions of miles, then a circle through the point (350, 370) just encloses Jupiter s orbit. Imagine replacing the sun with the star KY Cgni, whose radius is about 650 million miles. Would KY Cgni contain Jupiter s orbit? Eplain. (p. 66) 63 EXTRA PRACTICE for Lesson 9.3, p. 08 ONLINE QUIZ at classzone.com
Graphing Calculator ACTIVITY Use after Lesson 9.3 9.3 Graph Equations of Circles TEXAS classzone.com Kestrokes TEKS a.5, a.6, A.3.B QUESTION How can ou use a graphing calculator to graph a circle? To graph a circle on most graphing calculators, ou must first rewrite the circle s equation as two functions that taken together represent the circle. E XAMPLE Graph a circle Use a graphing calculator to graph 5 5. STEP Solve for Begin b solving the equation for. 5 5 5 5 56Ï } 5 Together, the functions 5 Ï } 5 and 5Ï } 5 represent the circle. STEP 3 Graph functions The graphs are shown in the standard window (0 0 and 0 0). Because the calculator screen is not square, a horizontal distance of unit is longer than a vertical distance of unit, and the circle is stretched into an oval. STEP Enter functions Enter the two functions as and. You can enter as. Y= (5X) Y=-Y Y3= Y4= Y5= Y6= Y7= STEP 4 Adjust graph To show the circle in true proportion, set a window so that the ratio of (Xma Xmin) to (Yma Ymin) is 3 :. Such a square window can also be obtained b pressing and selecting ZSquare. P RACTICE Use a graphing calculator to graph the equation. Give the viewing window that ou used and verif that it is a square window.. 5 44. 5 80 3. 5 576 4. 0.5 0.5 5 5. 7 7 5 05 6. 6 6 5 9 9.3 Graph and Write Equations of Circles 633