# Translate and Classify Conic Sections

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1 TEKS 9.6 A.5.A, A.5.B, A.5.D, A.5.E Trnslte nd Clssif Conic Sections Before You grphed nd wrote equtions of conic sections. Now You will trnslte conic sections. Wh? So ou cn model motion, s in E. 49. Ke Voculr conic sections (conics) generl seconddegree eqution discriminnt Becuse prols, circles, ellipses, nd hperols re formed when plne intersects doule-npped cone, the re clled conic sections or conics. Previousl, ou studied equtions of prols with vertices t the origin nd equtions of circles, ellipses, nd hperols with centers t the origin. Now ou will stud how trnslting conics in the coordinte plne ffects their equtions. KEY CONCEPT For Your Noteook Stndrd Form of Equtions of Trnslted Conics In the following equtions, the point (h, k) is the verte of the prol nd the center of the other conics. Circle ( h) ( k) 5 r Horizontl is Verticl is Prol ( k) 5 4p( h) ( h) 5 4p( k) Ellipse ( h) ( k) 5 ( h) ( k) 5 Hperol ( h) ( k) 5 ( k) ( h) 5 E XAMPLE Grph the eqution of trnslted circle Grph ( ) ( 3) 5 9. STEP STEP Compre the given eqution to the stndrd form of n eqution of circle. You cn see tht the grph is circle with center t (h, k) 5 (, 3) nd rdius r 5 Ï Plot the center. Then plot severl points tht re ech 3 units from the center: ( 3, 3) 5 (5, 3) ( 3, 3) 5 (, 3) (, 3 3) 5 (, 0) (, 3 3) 5 (, 6) Drw circle through the points. (, 0) 5 (, 3) (5, 3) (, 3) (, 6) 650 Chpter 9 Qudrtic Reltions nd Conic Sections

2 E XAMPLE Grph the eqution of trnslted hperol SOLVE FOR Y To plot dditionl points on the hperol, solve for to otin ( ) Î. 9 Then mke tle of vlues. Grph STEP STEP ( 3) ( ) Compre the given eqution to the stndrd forms of equtions of hperols. The eqution s form tells ou tht the grph is hperol with verticl trnsverse is. The center is t (h, k) 5 (, 3). Becuse 5 4 nd 5 9, ou know tht 5 nd 5 3. Plot the center, vertices, nd foci. The vertices lie 5 units ove nd elow the center, t (, 5) nd (, ). Becuse c 5 5 3, the foci lie c 5 Ï 3 ø 3.6 units ove nd elow the center, t (, 6.6) nd (, 0.6). Drw the hperol. Drw rectngle centered t (, 3) tht is 5 4 units high nd 5 6 units wide. Drw the smptotes through the opposite corners of the rectngle. Then drw the hperol pssing through the vertices nd pproching the smptotes. t clsszone.com (, 6.6) (, 3) (, 5) (, ) (, 0.6) GUIDED PRACTICE for Emples nd Grph the eqution. Identif the importnt chrcteristics of the grph.. ( ) ( 3) 5 4. ( ) 5 8( 3) 3. ( 3) ( 4) ( ) ( ) E XAMPLE 3 Write n eqution of trnslted prol Write n eqution of the prol whose verte is t (, 3) nd whose focus is t (4, 3). STEP STEP Determine the form of the eqution. Begin mking rough sketch of the prol. Becuse the focus is to the left of the verte, the prol opens to the left, nd its eqution hs the form ( k) 5 4p( h) where p < 0. Identif h nd k. The verte is t (, 3), so h 5 nd k 5 3. Find p. The verte (, 3) nd focus (4, 3) oth lie on the line 5 3, so the distnce etween them is p 5 4 () 5, nd thus p 56. Becuse p < 0, it follows tht p 5, so 4p 58. c The stndrd form of the eqution is ( 3) 58( ). (4, 3) (, 3) 9.6 Trnslte nd Clssif Conic Sections 65

3 E XAMPLE 4 Write n eqution of trnslted ellipse FIND DISTANCE The co-vertices lie on verticl line through the center nd the foci lie on horizontl line through the center, so ou do not hve to use the distnce formul. Write n eqution of the ellipse with foci t (, ) nd (7, ) nd co-vertices t (4, 0) nd (4, 4). STEP STEP Determine the form of the eqution. First sketch the ellipse. The foci lie on the mjor is, so the is is horizontl. The eqution hs this form: ( h) ( k) 5 Identif h nd k finding the center, which is hlfw etween the foci (or the co-vertices). (h, k) 5 7, 5 (4, ) Find, the distnce etween co-verte nd the center (4, ), nd c, the distnce etween focus nd the center. Choose the co-verte (4, 4) nd the focus (, ): nd c STEP 4 Find. For n ellipse, 5 c , so 5 Ï 3. c The stndrd form of the eqution is ( 4) ( ) (4, 4) (, ) (7, ) (4, 0) E XAMPLE 5 Identif smmetries of conic sections Identif the line(s) of smmetr for ech conic section in Emples (, 3) (, 3) 5 3 For the circle in Emple, n For the hperol in Emple, line through the center (, 3) 5 nd 5 3 re lines of is line of smmetr. smmetr. 5 4 (, 3) (4, ) For the prol in Emple 3, For the ellipse in Emple 4, is line of smmetr. nd 5 re lines of smmetr. 65 Chpter 9 Qudrtic Reltions nd Conic Sections

4 GUIDED PRACTICE for Emples 3, 4, nd 5 Write n eqution of the conic section. 5. Prol with verte t (3, ) nd focus t (3, ) 6. Hperol with vertices t (7, 3) nd (, 3) nd foci t (9, 3) nd (, 3) Identif the line(s) of smmetr for the conic section. 7. ( 5) 5 8. ( 5) 5 8( ) ( ) ( ) 5 49 KEY CONCEPT For Your Noteook Clssifing Conics Using Their Equtions An conic cn e descried generl second-degree eqution in nd : A B C D E F 5 0. The epression B 4AC is the discriminnt of the eqution nd cn e used to identif the tpe of conic. Discriminnt B 4AC < 0, B 5 0, nd A 5 C B 4AC < 0 nd either B Þ 0 or A Þ C B 4AC 5 0 B 4AC > 0 Tpe of Conic Circle Ellipse Prol Hperol If B 5 0, ech is of the conic is horizontl or verticl. E XAMPLE 6 Clssif conic Clssif the conic given Then grph the eqution. Note tht A 5 4, B 5 0, nd C 5, so the vlue of the discriminnt is: B 4AC 5 0 4(4)() 56 COMPLETE THE SQUARE For help with completing the squre, see p. 84. Becuse B 4AC < 0 nd A Þ C, the conic is n ellipse. To grph the ellipse, first complete the squre in (4 8) 5 8 (, 3) 4( ) 5 8 4(? ) 5 8 4(? ) 4( ) 5 8 4() 4( ) 5 ( ) 5 3 ( 3, 0) (, 0) ( 3, 0) (, 3) From the eqution, ou cn see tht (h, k) 5 (, 0), 5 Ï 5 Ï 3, nd 5 Ï 3. Use these fcts to drw the ellipse. 9.6 Trnslte nd Clssif Conic Sections 653

5 E XAMPLE 7 TAKS REASONING: Multi-Step Prolem PHYSICAL SCIENCE In l eperiment, ou record imges of steel ll rolling pst mgnet. The eqution models the ll s pth. Wht is the shpe of the pth? Write n eqution for the pth in stndrd form. Mgnet Grph the eqution of the pth. STEP Identif the shpe. The eqution is generl second-degree eqution with A 5 6, B 5 0, nd C 59. Find the vlue of the discriminnt. B 4AC 5 0 4(6)(9) STEP Becuse B 4AC > 0, the shpe of the pth is hperol. Write n eqution. To write n eqution of the hperol, complete the squre in oth nd simultneousl AVOID ERRORS To complete the squre in two vriles, ou must dd quntit to or sutrct quntit from ech side for ech vrile. (6 96) (9 36) ( 6? ) 9( 4? ) (? ) 9(? ) 6( 6 9) 9( 4 4) (9) 9(4) 6( 3) 9( ) 5 44 ( 3) ( ) Grph the eqution. From the eqution, the trnsverse is is horizontl, (h, k) 5 (3, ), 5 Ï 9 5 3, nd 5 Ï The vertices re t (3 6, ), or (6, ) nd (0, ). Plot the center nd vertices. Then drw rectngle 5 6 units wide nd 5 8 units high centered t (3, ), drw the smptotes, nd drw the hperol. Notice tht the pth of the ll is modeled just the right-hnd rnch of the hperol. (0, ) (6, ) (3, ) GUIDED PRACTICE for Emples 6 nd 7 Clssif the conic section nd write its eqution in stndrd form. Then grph the eqution ASTRONOMY An steroid s pth is modeled where nd re in stronomicl units from the sun. Clssif the pth nd write its eqution in stndrd form. Then grph the eqution. 654 Chpter 9 Qudrtic Reltions nd Conic Sections

6 9.6 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 3, 9, nd 49 5 TAKS PRACTICE AND REASONING Es., 36, 45, 5, 5, 54, nd 55. VOCABULARY Eplin wh circles, ellipses, prols, nd hperols re clled conic sections.. WRITING Eplin how the discriminnt of generl second-degree eqution cn e used to identif wht conic the eqution represents. nd on pp for Es. 3 GRAPHING Grph the eqution. Identif the importnt chrcteristics of the grph. 3. ( 4) 58( ) 4. ( ) ( 7) ( 6) 5 ( ) 5 6. ( 4) ( 8) ( ) 5 4( 6) 0. ( ) ( ) 5 8. ( 5) ( ) ( ) ( 3) ( 4) TAKS REASONING Wht re the coordintes of the co-vertices of the ( 4) ( ) ellipse with eqution 5? 6 4 A (0, ), (8, ) B (8, ), (0, ) C (4, 3), (4, ) D (4, 3), (4, ) 3 nd 4 on pp for Es. 3 WRITING EQUATIONS Write n eqution of the conic section. 3. Circle with center t (5, ) nd rdius 6 4. Circle with center t (9, ) nd rdius 5. Prol with verte t (4, 3) nd focus t (, 3) 6. Prol with verte t (5, 3) nd directri Ellipse with vertices t (3, 4) nd (5, 4) nd foci t (, 4) nd (3, 4) 8. Ellipse with vertices t (, ) nd (, 9) nd co-vertices t (4, 5) nd (0, 5) 9. Hperol with vertices t (6, 3) nd (6, ) nd foci t (6, 6) nd (6, 4) 0. Hperol with vertices t (, 7) nd (7, 7) nd foci t (, 7) nd (9, 7). ERROR ANALYSIS Descrie nd correct the error in writing n eqution of the ellipse with vertices t (7, 3) nd (3, 3) nd co-vertices t (, 6) nd (, 0). Ais is horizontl; (h, k) 5 (, 3); 5 7 () 5 5; ; ( ) ( 3) Eqution: EXAMPLE 5 on p. 65 for Es. 7 LINES OF SYMMETRY Identif the line(s) of smmetr for the conic section.. ( 5) ( ) 5 3. ( 4) 5 6( 6) ( ) ( ) ( 5) ( 3) ( 3) 5 0( ) 7. ( ) ( ) Trnslte nd Clssif Conic Sections 655

7 EXAMPLE 6 on p. 653 for Es CLASSIFYING CONICS Use the discriminnt to clssif the conic section TAKS REASONING The eqution represents wht conic section? A Circle B Ellipse C Hperol D Prol 6 nd 7 on pp for Es CLASSIFYING AND GRAPHING Clssif the conic section nd write its eqution in stndrd form. Then grph the eqution TAKS REASONING Consider generl second-degree eqution where B 5 0. Eplin how ou cn clssif the eqution s grph without grphing or using the discriminnt. 46. REASONING In Chpter 8, ou grphed hperols with equtions of the form 5. Write 5 s generl second-degree eqution, nd use the discriminnt to show tht the grph is hperol. 47. CHALLENGE Find epressions in terms of c, h, nd k for the coordintes of the foci of hperol with verticl trnsverse is nd center (h, k). Then find equtions of the smptotes in terms of,, h, nd k. PROBLEM SOLVING 3 nd 4 on pp for E nd 7 on pp for Es ICE SKATING A figure skter prctices skting figure eights, which re formed etching two eternll tngent circles in the ice. Write equtions for the circles in figure eight if ech is 8 feet in dimeter, the circles intersect t the origin, nd the centers of the circles re on the -is. 49. JUMPING STILTS The lep of person wering jumping stilts is modeled where nd re in feet nd the origin mrks the strt of the lep. Write n eqution in stndrd form for the pth of the lep. How high nd how fr does the person jump? 50. SPACECRAFT A spcecrft uses Sturn s grvittionl force to slingshot round the plnet on the pth , where the origin represents Sturn s center nd nd re in hundreds of thousnds of kilometers. Wht is the shpe of the pth? Write n eqution in stndrd form for the pth. Then grph the eqution WORKED-OUT SOLUTIONS on p. WS 5 TAKS PRACTICE AND REASONING

8 5. TAKS REASONING You re in prk surfing the Internet on wireless connection. A hotel s wireless trnsmitter is locted 00 rds est nd 60 rds south of ou. It hs rnge of 50 rds. A cfé s trnsmitter is locted 80 rds west nd 70 rds south of ou. It hs rnge of 00 rds.. With our loction s the origin, write inequlities for circulr regions round the hotel nd cfé in which ou cn get wireless Internet ccess.. Grph the inequlities. Are ou in onl one region or in oth? Eplin. c. Eplin how to determine whether the regions overlp without grphing. 5. TAKS REASONING Tell wht conic section is formed in the sitution descried. Eplin our resoning.. To use new tue of culk for the first time, ou cut the cone-shped tip digonll s shown.. When ou shrpen pencil with flt sides, ech side intersects the cone-shped tip s shown. 53. CHALLENGE Adegenerte conic results when the intersection of plne with doule-npped cone is not prol, circle, ellipse, or hperol. Digrm Digrm Digrm 3. In Digrm, plne perpendiculr to the cone s is psses through the cone, intersecting it in circle whose rdius decreses nd then increses. When is the intersection not circle? Wht is it?. In Digrm, plne prllel to the cone s is psses through the cone, intersecting it in hperol whose vertices get closer together nd then frther prt. When is the intersection not hperol? Wht is it? c. In Digrm 3, plne prllel to the cone s nppe psses through the cone, intersecting it in prol tht first gets nrrower, then flips nd gets wider. When is the intersection not prol? Wht is it? MIXED REVIEW FOR TAKS TAKS PRACTICE t clsszone.com REVIEW Lesson.5; TAKS Workook 54. TAKS PRACTICE In 003, the popultion of Tes ws out 39,000 less thn 3 times the popultion of Virgini. Let represent the popultion of Virgini. Which epression represents the popultion of Tes? TAKS Oj. A 39,000 3 B 39,000 3 C 3 39,000 D 3 39,000 REVIEW TAKS Preprtion p. 408; TAKS Workook 55. TAKS PRACTICE The mesure of ech interior ngle of regulr polgon is 358. How mn sides does the polgon hve? TAKS Oj. 6 F 6 G 7 H 8 J 9 EXTRA PRACTICE for Lesson 9.6, p. 08 ONLINE QUIZ t clsszone.com 657

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