Write and Graph Equations of Circles
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1 0.7 Wite and Gaph Equations of icles Befoe You wote equations of lines in the coodinate plane. Now You will wite equations of cicles in the coodinate plane. Wh? So ou can detemine zones of a commute sstem, as in E. 36. Ke Vocabula standad equation of a cicle Let (, ) epesent an point on a cicle with cente at the oigin and adius. B the Pthagoean Theoem, This is the equation of a cicle with adius and cente at the oigin. (, ) E X A M P L E Wite an equation of a cicle Wite the equation of the cicle shown. The adius is 3 and the cente is at the oigin Equation of cicle Substitute Simplif. c The equation of the cicle is IRLES ENTERED AT (h, k) You can wite the equation of an cicle if ou know its adius and the coodinates of its cente. Suppose a cicle has adius and cente (h, k). Let (, ) be a point on the cicle. The distance between (, ) and (h, k) is, so b the Distance Fomula Ï }} ( 2 h) 2 ( 2 k) 2 5. Squae both sides to find the standad equation of a cicle. (h, k) (, ) KEY ONEPT Fo You Notebook Standad Equation of a icle The standad equation of a cicle with cente (h, k) and adius is: ( 2 h) 2 ( 2 k) Wite and Gaph Equations of icles 699
2 E X A M P L E 2 Wite the standad equation of a cicle Wite the standad equation of a cicle with cente (0, 29) and adius 4.2. ( 2 h) 2 ( 2 k) Standad equation of a cicle ( 2 0) 2 ( 2 (29)) Substitute. 2 ( 9) Simplif. GUIDED PRATIE fo Eamples and 2 Wite the standad equation of the cicle with the given cente and adius.. ente (0, 0), adius ente (22, 5), adius 7 E X A M P L E 3 Wite the standad equation of a cicle The point (25, 6) is on a cicle with cente (2, 3). Wite the standad equation of the cicle. (25, 6) To wite the standad equation, ou need to know the values of h, k, and. To find, find the distance between the cente and the point (25, 6) on the cicle. (2, 3) 5 Ï }}} [25 2 (2)] 2 (6 2 3) 2 Distance Fomula 5 Ï } (24) Simplif. 5 5 Simplif. Substitute (h, k) 5 (2, 3) and 5 5 into the standad equation of a cicle. ( 2 h) 2 ( 2 k) Standad equation of a cicle [ 2 (2)] 2 ( 2 3) Substitute. ( ) 2 ( 2 3) Simplif. c The standad equation of the cicle is ( ) 2 ( 2 3) GUIDED PRATIE fo Eample 3 3. The point (3, 4) is on a cicle whose cente is (, 4). Wite the standad equation of the cicle. 4. The point (2, 2) is on a cicle whose cente is (2, 6). Wite the standad equation of the cicle. 700 hapte 0 Popeties of icles
3 E X A M P L E 4 Gaph a cicle USE EQUATIONS If ou know the equation of a cicle, ou can gaph the cicle b identifing its cente and adius. The equation of a cicle is ( 2 4) 2 ( 2) Gaph the cicle. Rewite the equation to find the cente and adius. ( 2 4) 2 ( 2) ( 2 4) 2 [ 2 (22)] The cente is (4, 22) and the adius is 6. Use a compass to gaph the cicle. 4 2 (4, 22) E X A M P L E 5 Use gaphs of cicles EARTHQUAKES The epicente of an eathquake is the point on Eath s suface diectl above the eathquake s oigin. A seismogaph can be used to detemine the distance to the epicente of an eathquake. Seismogaphs ae needed in thee diffeent places to locate an eathquake s epicente. Use the seismogaph eadings fom locations A, B, and to find the epicente of an eathquake. The epicente is 7 miles awa fom A(22, 2.5). The epicente is 4 miles awa fom B(4, 6). The epicente is 5 miles awa fom (3, 22.5). The set of all points equidistant fom a given point is a cicle, so the epicente is located on each of the following cicles. (A with cente (22, 2.5) and adius 7 (B with cente (4, 6) and adius 4 A 8 4 B ( with cente (3, 22.5) and adius 5 To find the epicente, gaph the cicles on a gaph whee units ae measued in miles. Find the point of intesection of all thee cicles c The epicente is at about (5, 2). at classzone.com GUIDED PRATIE fo Eamples 4 and 5 5. The equation of a cicle is ( 2 4) 2 ( 3) Gaph the cicle. 6. The equation of a cicle is ( 8) 2 ( 5) Gaph the cicle. 7. Wh ae thee seismogaphs needed to locate an eathquake s epicente? 0.7 Wite and Gaph Equations of icles 70
4 0.7 EXERISES SKILL PRATIE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS fo Es. 7, 7, and 37 5 STANDARDIZED TEST PRATIE Es. 2, 6, 26, and 42. VOABULARY op and complete: The standad equation of a cicle can be witten fo an cicle with known? and?. 2. WRITING Eplain wh the location of the cente and one point on a cicle is enough infomation to daw the est of the cicle. EXAMPLES and 2 on pp fo Es. 3 6 WRITING EQUATIONS Wite the standad equation of the cicle WRITING EQUATIONS Wite the standad equation of the cicle with the given cente and adius. 9. ente (0, 0), adius 7 0. ente (24, ), adius. ente (7, 26), adius 8 2. ente (4, ), adius 5 3. ente (3, 25), adius 7 4. ente (23, 4), adius 5 5. ERROR ANALYSIS Descibe and coect the eo in witing the equation of a cicle. An equation of a cicle with cente (23, 25) and adius 3 is ( 2 3) 2 ( 2 5) MULTIPLE HOIE The standad equation of a cicle is ( 2 2) 2 ( ) What is the diamete of the cicle? A 2 B 4 8 D 6 EXAMPLE 3 on p. 700 fo Es. 7 9 WRITING EQUATIONS Use the given infomation to wite the standad equation of the cicle. 7. The cente is (0, 0), and a point on the cicle is (0, 6). 8. The cente is (, 2), and a point on the cicle is (4, 2). 9. The cente is (23, 5), and a point on the cicle is (, 8). 702 hapte 0 Popeties of icles
5 EXAMPLE 4 on p. 70 fo Es GRAPHING IRLES Gaph the equation ( 2 3) ( 2) ( 2 4) 2 ( 2 ) ( 5) 2 ( 2 3) ( 2) 2 ( 6) MULTIPLE HOIE Which of the points does not lie on the cicle descibed b the equation ( 2) 2 ( 2 4) ? A (22, 2) B (, 8) (3, 4) D (0, 5) ALGEBRA Detemine whethe the given equation defines a cicle. If the equation defines a cicle, ewite the equation in standad fom IDENTIFYING TYPES OF LINES Use the given equations of a cicle and a line to detemine whethe the line is a tangent, secant, secant that contains a diamete, o none of these. 3. icle: ( 2 4) 2 ( 2 3) icle: ( 2) 2 ( 2 2) Line: Line: icle: ( 2 5) 2 ( ) icle: ( 3) 2 ( 2 6) Line: 5 } Line: } HALLENGE Fou tangent cicles ae centeed on the -ais. The adius of (A is twice the adius of (O. The adius of (B is thee times the adius of (O. The adius of ( is fou times the adius of (O. All cicles have intege adii and the point (63, 6) is on (. What is the equation of (A? O A B PROBLEM SOLVING EXAMPLE 5 on p. 70 fo E OMMUTER TRAINS A cit s commute sstem has thee zones coveing the egions descibed. Zone coves people living within thee miles of the cit cente. Zone 2 coves those between thee and seven miles fom the cente, and Zone 3 coves those ove seven miles fom the cente. a. Gaph this situation with the cit cente at the oigin, whee units ae measued in miles. b. Find which zone coves people living at (3, 4), (6, 5), (, 2), (0, 3), and (, 6). Zone 3 Zone mi 40 Zone Wite and Gaph Equations of icles 703
6 37. OMPAT DISS The diamete of a D is about 4.8 inches. The diamete of the hole in the cente is about 0.6 inches. You place a D on the coodinate plane with cente at (0, 0). Wite the equations fo the outside edge of the disc and the edge of the hole in the cente. 4.8 in. 0.6 in. REULEAUX POLYGONS In Eecises 38 4, use the following infomation. The figue at the ight is called a Reuleau polgon. It is not a tue polgon because its sides ae not staight. n AB is equilateal. 38. JD lies on a cicle with cente A and adius AD. Wite an equation of this cicle. 39. DE lies on a cicle with cente B and adius BD. Wite an equation of this cicle. E F 40. ONSTRUTION The emaining acs of the polgon ae constucted in the same wa as JD and DE in Eecises 38 and 39. onstuct a Reuleau polgon on a piece of cadboad. 4. ut out the Reuleau polgon fom Eecise 40. Roll it on its edge like a wheel and measue its height when it is in diffeent oientations. Eplain wh a Reuleau polgon is said to have constant width. 42. EXTENDED RESPONSE Telecommunication towes can be used to tansmit cellula phone calls. Towes have a ange of about 3 km. A gaph with units measued in kilometes shows towes at points (0, 0), (0, 5), and (6, 3). a. Daw the gaph and locate the towes. Ae thee an aeas that ma eceive calls fom moe than one towe? b. Suppose ou home is located at (2, 6) and ou school is at (2.5, 3). an ou use ou cell phone at eithe o both of these locations? c. it A is located at (22, 2.5) and it B is at (5, 4). Each cit has a adius of.5 km. Which cit seems to have bette cell phone coveage? Eplain. D A J B H G 43. REASONING The lines 5 } 3 2 and 5 2} ae tangent to ( at the points (4, 5) and (4, 3), espectivel. a. Find the coodinates of and the adius of (. Eplain ou steps. b. Wite the standad equation of ( and daw its gaph. 44. PROOF Wite a poof. GIVEN c A cicle passing though the points (2, 0) and (, 0) PROVE c The equation of the cicle is 2 2 2k 2 5 with cente at (0, k). (2, 0) (, 0) WORKED-OUT SOLUTIONS on p. WS 5 STANDARDIZED TEST PRATIE
7 45. HALLENGE The intesecting lines m and n ae tangent to ( at the points (8, 6) and (0, 8), espectivel. a. What is the intesection point of m and n if the adius of ( is 2? What is thei intesection point if is 0? What do ou notice about the two intesection points and the cente? b. Wite the equation that descibes the locus of intesection points of m and n fo all possible values of. MIXED REVIEW PREVIEW Pepae fo Lesson. in Es Find the peimete of the figue. 46. (p. 49) 47. (p. 49) 48. (p. 433) 9 in. 8 ft 40 m 22 in. 57 m Find the cicumfeence of the cicle with given adius o diamete d. Use p (p. 49) cm 50. d 5 60 in. 5. d 5 48 d Find the adius of (. (p. 65) QUIZ fo Lessons Find the value of. (p. 689) In Eecises 4 and 5, use the given infomation to wite the standad equation of the cicle. (p. 699) 4. The cente is (, 4), and the adius is The cente is (5, 27), and a point on the cicle is (5, 23). 6. TIRES The diamete of a cetain tie is 24.2 inches. The diamete of the im in the cente is 4 inches. Daw the tie in a coodinate plane with cente at (24, 3). Wite the equations fo the oute edge of the tie and fo the im whee units ae measued in inches. (p. 699) EXTRA PRATIE fo Lesson 0.7, p. 95 ONLINE QUIZ at classzone.com 705
8 MIXED REVIEW of Poblem Solving Lessons SHORT RESPONSE A local adio station can boadcast its signal 20 miles. The station is located at the point (20, 30) whee units ae measued in miles. a. Wite an inequalit that epesents the aea coveed b the adio station. b. Detemine whethe ou can eceive the adio station s signal when ou ae located at each of the following points: E(25, 25), F(0, 0), G(20, 6), and H(35, 30). 4. SHORT RESPONSE You ae at point A, about 6 feet fom a cicula aquaium tank. The distance fom ou to a point of tangenc on the tank is 7 feet. 7 ft B A 6 ft ft ft STATE TEST PRATIE classzone.com 2. EXTENDED RESPONSE ell phone towes ae used to tansmit calls. An aea has cell phone towes at points (2, 3), (4, 5), and (5, 3) whee units ae measued in miles. Each towe has a tansmission adius of 2 miles. a. Daw the aea on a gaph and locate the thee cell phone towes. Ae thee an aeas that can tansmit calls using moe than one towe? b. Suppose ou live at (3, 5) and ou fiend lives at (, 7). an ou use ou cell phone at eithe o both of ou homes? c. it A is located at (2, ) and it B is located at (4, 7). Each cit has a adius of 5 miles. Which cit has bette coveage fom the cell phone towes? 3. SHORT RESPONSE You ae standing at point P inside a go-kat tack. To detemine if the tack is a cicle, ou measue the distance to fou points on the tack, as shown in the diagam. What can ou conclude about the shape of the tack? Eplain. a. What is the adius of the tank? b. Suppose ou ae standing 4 feet fom anothe aquaium tank that has a diamete of 2 feet. How fa, in feet, ae ou fom a point of tangenc? 5. EXTENDED RESPONSE You ae given seismogaph eadings fom thee locations. At A(22, 3), the epicente is 4 miles awa. At B(5, 2), the epicente is 5 miles awa. At (2, 5), the epicente is 2 miles awa. a. Gaph cicles centeed at A, B, and with adii of 4, 5, and 2 miles, espectivel. b. Locate the epicente. c. The eathquake could be felt up to 2 miles awa. If ou live at (4, 6), could ou feel the eathquake? Eplain. 6. MULTI-STEP PROBLEM Use the diagam. 5 8 a. Use Theoem 0.6 and the quadatic fomula to wite an equation fo in tems of. b. Find the value of. c. Find the value of. 706 hapte 0 Popeties of icles
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