SECTION 11-4 Translation of Axes

Transcription

1 11- Trnsltion of Aes 811 Rdiotelescope For the receiving ntenn shown in the figure, the common focus F is locted 10 feet bove the verte of the prbol, nd focus F (for the hperbol) is 0 feet bove the verte. The verte of the reflecting hperbol is 110 feet bove the verte for the prbol. Introduce coordinte sstem b using the is of the prbol s the is (up positive), nd let the is pss through the center of the hperbol (right positive). Wht is the eqution of the reflecting hperbol? Write in terms of. (b) SECTION 11- Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A C D E F 0 Finding Equtions of Conics In the lst three sections we found stndrd equtions for prbols, ellipses, nd hperbols locted with their es on the coordinte es nd centered reltive to the origin. Wht hppens if we move conics w from the origin while keeping their es prllel to the coordinte es? We will show tht we cn obtin new stndrd equtions tht re specil cses of the eqution A C D E F 0, where A nd C re not both zero. The bsic mthemticl tool used in this endevor is trnsltion of es. The usefulness of trnsltion of es is not limited to grphing conics, however. Trnsltion of es cn be put to good use in mn other grphing situtions. Trnsltion of Aes A trnsltion of coordinte es occurs when the new coordinte es hve the sme direction s nd re prllel to the originl coordinte es. To see how coordintes in the originl sstem re chnged when moving to the trnslted sstem, nd vice vers, refer to Figure 1. FIGURE 1 Trnsltion of coordintes. P(, ) P(, ) 0 (0, 0) (0, 0) (h, k) 0

2 81 11 Additionl Topics in Anltic Geometr A point P in the plne hs two sets of coordintes: (, ) in the originl sstem nd (, ) in the trnslted sstem. If the coordintes of the origin of the trnslted sstem re (h, k) reltive to the originl sstem, then the old nd new coordintes re relted s given in Theorem 1. Theorem 1 Trnsltion Formuls 1. h. h k k It cn be shown tht these formuls hold for (h, k) locted nwhere in the originl coordinte sstem. EXAMPLE 1 Eqution of Curve in Trnslted Sstem A curve hs the eqution ( ) ( 1) 36 If the origin is trnslted to (, 1), find the eqution of the curve in the trnslted sstem nd identif the curve. Solution Since (h, k) (, 1), use trnsltion formuls h k 1 to obtin, fter substitution, 36 This is the eqution of circle of rdius 6 with center t the new origin. The coordintes of the new origin in the originl coordinte sstem re (, 1) (Fig. ). Note tht this result grees with our generl tretment of the circle in Section -1. FIGURE ( ) ( 1) A(, 1) 10

3 11- Trnsltion of Aes 813 Mtched Problem 1 A curve hs the eqution ( ) 8( 3). If the origin is trnslted to (3, ), find n eqution of the curve in the trnslted sstem nd identif the curve. Stndrd Equtions of Trnslted Conics We now proceed to find stndrd equtions of conics trnslted w from the origin. We do this b first writing the stndrd equtions found in erlier sections in the coordinte sstem with 0 t (h, k). We then use trnsltion equtions to find the stndrd forms reltive to the originl coordinte sstem. The equtions of trnsltion in ll cses re h k For prbols we hve ( h) ( k) ( k) ( h) For circles we hve For ellipses we hve for b 0 r ( h) ( k) r For hperbols we hve b 1 ( h) ( k) 1 b b 1 ( h) ( k) 1 b b 1 ( h) ( k) 1 b b 1 ( k) ( h) 1 b Tble 1 summrizes these results with pproprite figures nd some properties discussed erlier.

4 81 11 Additionl Topics in Anltic Geometr TABLE 1 Stndrd Equtions for Trnslted Conics Prbols ( h) ( k) ( k) ( h) F V(h, k) Verte (h, k) Focus (h, k ) 0 opens up 0 opens down V(h, k) F Verte (h, k) Focus (h, k) 0 opens left 0 opens right Circles ( h) ( k) r Center (h, k) Rdius r r C(h, k) ( h) ( k) 1 b Ellipses b 0 ( h) ( k) 1 b b (h, k) Center (h, k) Mjor is Minor is b Center (h, k) Mjor is Minor is b (h, k) b (h, k) ( h) ( k) 1 b b Center (h, k) Trnsverse is Conjugte is b Hperbols (h, k) ( k) ( h) 1 b b Center (h, k) Trnsverse is Conjugte is b

5 11- Trnsltion of Aes 81 Grphing Equtions of the Form A C D E F 0 It cn be shown tht the grph of A C D E F 0 (1) where A nd C re not both zero, is conic or degenerte conic or tht there is no grph. If we cn trnsform eqution (1) into one of the stndrd forms in Tble 1, then we will be ble to identif its grph nd sketch it rther quickl. The process of completing the squre discussed in Section 1-6 will be our primr tool in ccomplishing this trnsformtion. A couple of emples should help mke the process cler. EXAMPLE Grphing Trnslted Conic Trnsform () into one of the stndrd forms in Tble 1. Identif the conic nd grph it. Solution Step 1. Complete the squre in eqution () reltive to ech vrible tht is squred in this cse : Add 9 to both sides to complete the squre on the left side. ( 3) ( ) (3) From Tble 1 we recognize eqution (3) s n eqution of prbol opening to the right with verte t (h, k) (, 3). Step. Find the eqution of the prbol in the trnslted sstem with origin 0 t (h, k) (, 3). The equtions of trnsltion re red directl from eqution (3): 3 A(, 3) 0 Mking these substitutions in eqution (3) we obtin () the eqution of the prbol in the sstem. Step 3. Grph eqution () in the sstem following the process discussed in Section The resulting grph is the grph of the originl eqution reltive to the originl coordinte sstem (Fig. 3). FIGURE

6 Additionl Topics in Anltic Geometr Mtched Problem Trnsform 1 0 into one of the stndrd forms in Tble 1. Identif the conic nd grph it. EXAMPLE 3 Grphing Trnslted Conic Trnsform into one of the stndrd forms in Tble 1. Identif the conic nd grph it. Find the coordintes of n foci reltive to the originl sstem. Solution Step 1. Complete the squre reltive to both nd ) 36 9( ) ( 6 9) 36 9( ) ( 6 9) ( ) ( 3) 36 ( ) From Tble 1 we recognize the lst eqution s n eqution of hperbol opening left nd right with center t (h, k) (, 3). Step. Find the eqution of the hperbol in the trnslted sstem with origin 0 t (h, k) (, 3). The equtions of trnsltion re red directl from the lst eqution in step 1: 3 Mking these substitutions, we obtin ( 3) the eqution of the hperbol in the sstem. Step 3. Grph the eqution obtined in step in the sstem following the process discussed in Section The resulting grph is the grph of the originl eqution reltive to the originl coordinte sstem (Fig. ). 1

7 11- Trnsltion of Aes 817 FIGURE F c F c 10 Step. Find the coordintes of the foci. To find the coordintes of the foci in the originl sstem, first find the coordintes in the trnslted sstem: Thus, the coordintes in the trnslted sstem re Now, use to obtin c 3 13 c 13 c 13 F( 13, 0) nd F( 13, 0) h k 3 F( 13, 3) nd F( 13, 3) s the coordintes of the foci in the originl sstem. Mtched Problem 3 Trnsform into one of the stndrd forms in Tble 1. Identif the conic nd grph it. Find the coordintes of n foci reltive to the originl sstem. Remrk. A grphing utilit provides n lterntive pproch to grphing equtions of the form A C D E F 0. Consider, for emple, the eqution of Emple 3. We write the eqution s qudrtic eqution in the vrible : ( ) 0. B the

8 Additionl Topics in Anltic Geometr qudrtic formul, 16f(), where f() We then grph ech of the two functions in the epression for. The grph of 16f() is the upper hlf of the hperbol, nd the grph of 8 16f() is the lower hlf. 8 D E EXPLORE-DISCUSS 1 If A 0 nd C 0, show tht the trnsltion of es, A C trnsforms the eqution A C D E F 0 into n eqution of the form A C K. Finding Equtions of Conics We now reverse the problem: Given certin informtion bout conic in rectngulr coordinte sstem, find its eqution. EXAMPLE Finding the Eqution of Trnslted Conic Find the eqution of hperbol with vertices on the line, conjugte is on the line 3, length of the trnsverse is, nd length of the conjugte is 6. Solution Locte the vertices, smptote rectngle, nd smptotes in the originl coordinte sstem [Fig. ()], then sketch the hperbol nd trnslte the origin to the center of the hperbol [Fig. (b)]. FIGURE b 3 3 () Asmptote rectngle (b) Hperbol Net write the eqution of the hperbol in the trnslted sstem: 9 1

9 11- Trnsltion of Aes 819 The origin in the trnslted sstem is t (h, k) (, 3), nd the trnsltion formuls re h () k 3 Thus, the eqution of the hperbol in the originl sstem is ( 3) ( ) 9 1 or, fter simplifing nd writing in the form of eqution (1), Mtched Problem Find the eqution of n ellipse with foci on the line, minor is on the line 3, length of the mjor is 8, nd length of the minor is. EXPLORE-DISCUSS Use the strteg of completing the squre to trnsform ech eqution to n eqution in n coordinte sstem. Note tht the eqution ou obtin is not one of the stndrd forms in Tble 1; insted, it is either the eqution of degenerte conic or the eqution hs no solution. If the solution set of the eqution is not empt, grph it nd identif the grph ( point, line, two prllel lines, or two intersecting lines). (A) (B) 3 0 (C) 1 0 (D) (E) Answers to Mtched Problems 1. 8; prbol. ( ) ( ); prbol (, )

10 80 11 Additionl Topics in Anltic Geometr ( ) ( 1) 3. 1; ellipse Foci: F( 7, 1), F( 7, 1) 16 9 F F ( ) ( 3). 1, or EXERCISE 11- A In Problems 1 8: (A) Find trnsltion formuls tht trnslte the origin to the indicted point (h, k). (B) Write the eqution of the curve for the trnslted sstem. (C) Identif the curve. 1. ( 3) ( ) 81; (3, ). ( 3) 8( ); (3, ) ( 7) ( ) 3. 1; (7, ) ( ) ( 6) 36; (, 6). ( 9) 16( ); (, 9) ( 9) ( ) 6. 1; (, 9) 10 6 ( 8) ( 3) 7. 1; (8, 3) 1 8 ( 7) ( 8) 8. 1; (7, 8) 0 In Problems 9 1: (A) Write ech eqution in one of the stndrd forms listed in Tble 1. (B) Identif the curve ( 3) 9( ) ( ) 1( 3) ( ) ( 7) ( ) 8( 3) 13. ( 6) ( ) 0 1. ( 7) 7( 3) 8 B In Problems 1, trnsform ech eqution into one of the stndrd forms in Tble 1. Identif the curve nd grph it If A 0, C 0, nd E 0, find h nd k so tht the trnsltion of es h, k trnsforms the eqution A C D E F 0 into one of the stndrd forms of Tble 1.. If A 0, C 0, nd D 0, find h nd k so tht the trnsltion of es h, k trnsforms the eqution A C D E F 0 into one of the stndrd forms of Tble 1. In Problems 3, use the given informtion to find the eqution of ech conic. Epress the nswer in the form A C D E F 0 with integer coefficients nd A 0.. A prbol with verte t (, 3), nd focus t (, 11). 6. A prbol with focus t (, 3), nd directri the is. 7. An ellipse with vertices (3, ) nd (3, 10) nd length of minor is A hperbol with vertices (, 8) nd (, 8) nd length of conjugte is.

11 11- Prmetric Equtions A hperbol with foci (, 1) nd (6, 1) nd vertices (3, 1) nd (, 1). 30. An ellipse with foci (3, 0) nd (3, 6) nd vertices (3, ) nd (3, 8). 31. A prbol with is the is nd pssing through the points (1, 0) nd (, ). 3. A prbol with verte t (6, ), is the line, nd pssing through the point (0, 7). 33. An ellipse with vertices (1, 1), nd (, 1) tht psses through the origin. 3. A hperbol with vertices t (, 3), nd (, ) tht psses through the point (, 0). C In Problems 3 0, find the coordintes of n foci reltive to the originl coordinte sstem: 3. Problem Problem Problem Problem Problem 1 0. Problem In Problems 1, use grphing utilit to find the coordintes of ll points of intersection to two deciml plces , , , , SECTION 11- Prmetric Equtions Prmetric Equtions nd Plne Curves Projectile Motion Ccloid Prmetric Equtions nd Plne Curves Consider the two equtions t 1 t t t (1) Ech vlue of t determines vlue of, vlue of, nd hence, n ordered pir (, ). To grph the set of ordered pirs (, ) determined b letting t ssume ll rel vlues, we construct Tble 1 listing selected vlues of t nd the corresponding vlues of nd. Then we plot the ordered pirs (, ) nd connect them with continuous curve, s shown in Figure 1. The vrible t is clled prmeter nd does not pper on the grph. Equtions (1) re clled prmetric equtions becuse both nd re epressed in terms of the prmeter t. The grph of the ordered pirs (, ) is clled plne curve. TABLE 1 10 t FIGURE 1 Grph of t 1, t t, t. In some cses it is possible to eliminte the prmeter b solving one of the equtions for t nd substituting into the other. In the emple just considered, solving the first eqution for t in terms of, we hve t 1