SECTION 94 Translation of Axes


 Chester Cooper
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1 94 Trnsltion of Aes 639 Rdiotelescope For the receiving ntenn shown in the figure, the common focus F is locted 120 feet bove the verte of the prbol, nd focus F (for the hperbol) is 20 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 94 Trnsltion of Aes Trnsltion of Aes Stndrd Equtions of Trnslted Conics Grphing Equtions of the Form A 2 C 2 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 2 C 2 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
2 640 9 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 2. 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 ( 4) 2 ( 1) 2 36 If the origin is trnslted to (4, 1), find the eqution of the curve in the trnslted sstem nd identif the curve. Solution Since (h, k) (4, 1), use trnsltion formuls h 4 k 1 to obtin, fter substitution, 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 (4, 1) (Fig. 2). Note tht this result grees with our generl tretment of the circle in Section 31. FIGURE 2 ( 4) 2 ( 1) A(4, 1) 10
3 94 Trnsltion of Aes 641 Mtched Problem 1 A curve hs the eqution ( 2) 2 8( 3). If the origin is trnslted to (3, 2), 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 2 4 ( h) 2 4( k) 2 4 ( k) 2 4( h) For circles we hve For ellipses we hve for b r 2 ( h) 2 ( k) 2 r 2 For hperbols we hve b 1 ( h) 2 ( k)2 2 2 b b 1 ( h) 2 ( k)2 2 b b 1 ( h) 2 ( k)2 2 2 b b 1 ( k) 2 ( h)2 2 2 b 2 Tble 1 summrizes these results with pproprite figures nd some properties discussed erlier.
4 642 9 Additionl Topics in Anltic Geometr TABLE 1 Stndrd Equtions for Trnslted Conics Prbols ( h) 2 4( k) ( k) 2 4( 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) 2 ( k) 2 r 2 Center (h, k) Rdius r r C(h, k) ( h) 2 ( k)2 2 b 2 Ellipses b 0 ( h) 2 ( k)2 b 2 2 b (h, k) Center (h, k) Mjor is 2 Minor is 2b Center (h, k) Mjor is 2 Minor is 2b (h, k) b (h, k) ( h) 2 ( k)2 2 b 2 b Center (h, k) Trnsverse is 2 Conjugte is 2b Hperbols (h, k) ( k) 2 ( h)2 2 b 2 b Center (h, k) Trnsverse is 2 Conjugte is 2b
5 94 Trnsltion of Aes 643 Grphing Equtions of the Form A 2 C 2 D E F 0 It cn be shown tht the grph of A 2 C 2 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 26 will be our primr tool in ccomplishing this trnsformtion. A couple of emples should help mke the process cler. EXAMPLE 2 Grphing Trnslted Conic Trnsform (2) into one of the stndrd forms in Tble 1. Identif the conic nd grph it. Solution Step 1. Complete the squre in eqution (2) reltive to ech vrible tht is squred in this cse : Add 9 to both sides to complete the squre on the left side. ( 3) 2 4( 2) (3) From Tble 1 we recognize eqution (3) s n eqution of prbol opening to the right with verte t (h, k) (2, 3). Step 2. Find the eqution of the prbol in the trnslted sstem with origin 0 t (h, k) (2, 3). The equtions of trnsltion re red directl from eqution (3): 2 3 A(2, 3) 0 Mking these substitutions in eqution (3) we obtin 2 4 (4) the eqution of the prbol in the sstem. Step 3. Grph eqution (4) in the sstem following the process discussed in Section 91. The resulting grph is the grph of the originl eqution reltive to the originl coordinte sstem (Fig. 3). FIGURE
6 644 9 Additionl Topics in Anltic Geometr Mtched Problem 2 Trnsform 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 )4 36 9( 2 4 4) 4( 2 6 9) 36 9( 2 4 4) 4( 2 6 9) ( 2) 2 4( 3) 2 36 ( 2) 2 4 From Tble 1 we recognize the lst eqution s n eqution of hperbol opening left nd right with center t (h, k) (2, 3). Step 2. Find the eqution of the hperbol in the trnslted sstem with origin 0 t (h, k) (2, 3). The equtions of trnsltion re red directl from the lst eqution in step 1: 2 3 Mking these substitutions, we obtin 2 4 ( 3) the eqution of the hperbol in the sstem. Step 3. Grph the eqution obtined in step 2 in the sstem following the process discussed in Section 93. The resulting grph is the grph of the originl eqution reltive to the originl coordinte sstem (Fig. 4). 1
7 94 Trnsltion of Aes 64 FIGURE F c F c 10 Step 4. 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 c 13 c 13 F( 13, 0) nd F( 13, 0) h 2 k 3 F( 13 2, 3) nd F( 13 2, 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 2 C 2 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
8 646 9 Additionl Topics in Anltic Geometr qudrtic formul, f(), where f() We then grph ech of the two functions in the epression for. The grph of f() is the upper hlf of the hperbol, nd the grph of f() is the lower hlf. 8 D E EXPLOREDISCUSS 1 If A 0 nd C 0, show tht the trnsltion of es, 2A 2C trnsforms the eqution A 2 C 2 D E F 0 into n eqution of the form A 2 C 2 K. Finding Equtions of Conics We now reverse the problem: Given certin informtion bout conic in rectngulr coordinte sstem, find its eqution. EXAMPLE 4 Finding the Eqution of Trnslted Conic Find the eqution of hperbol with vertices on the line 4, conjugte is on the line 3, length of the trnsverse is 4, 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 4 2 b 3 3 () Asmptote rectngle (b) Hperbol Net write the eqution of the hperbol in the trnslted sstem:
9 94 Trnsltion of Aes 647 The origin in the trnslted sstem is t (h, k) (4, 3), nd the trnsltion formuls re h (4) 4 k 3 Thus, the eqution of the hperbol in the originl sstem is ( 3) 2 4 ( 4)2 9 1 or, fter simplifing nd writing in the form of eqution (1), Mtched Problem 4 Find the eqution of n ellipse with foci on the line 4, minor is on the line 3, length of the mjor is 8, nd length of the minor is 4. EXPLOREDISCUSS 2 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) (C) (D) (E) Answers to Mtched Problems ; prbol 2. ( 2) 2 4( 4); prbol (2, 4)
10 648 9 Additionl Topics in Anltic Geometr ( 2) 2 ( 1)2 3. ; ellipse Foci: F( 7 2, 1), F( 7 2, 1) 16 9 F F ( 4) 2 ( 3)2 4., or EXERCISE 94 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) 2 ( ) 2 81; (3, ) 2. ( 3) 2 8( 2); (3, 2) ( 7) 2 ( 4)2 3. ; (7, 4) ( 2) 2 ( 6) 2 36; (2, 6). ( 9) 2 16( 4); (4, 9) ( 9) 2 ( )2 6. ; (, 9) 10 6 ( 8) 2 ( 3)2 7. ; (8, 3) 12 8 ( 7) 2 ( 8)2 8. ; (7, 8) 2 0 In Problems 9 14: (A) Write ech eqution in one of the stndrd forms listed in Tble 1. (B) Identif the curve ( 3) 2 9( 2) ( 2) 2 12( 3) ( ) 2 ( 7) ( ) 2 8( 3) ( 6) 2 24( 4) ( 7) 2 7( 3) 2 28 B In Problems 1 22, 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 2 C 2 D E F 0 into one of the stndrd forms of Tble If A 0, C 0, nd D 0, find h nd k so tht the trnsltion of es h, k trnsforms the eqution A 2 C 2 D E F 0 into one of the stndrd forms of Tble 1. In Problems 2 34, use the given informtion to find the eqution of ech conic. Epress the nswer in the form A 2 C 2 D E F 0 with integer coefficients nd A A prbol with verte t (, 3), nd focus t (, 11). 26. A prbol with focus t (2, 3), nd directri the is. 27. An ellipse with vertices (3, 2) nd (3, 10) nd length of minor is A hperbol with vertices (2, 8) nd (4, 8) nd length of conjugte is 24.
11 Chpter 9 Group Activit A hperbol with foci (2, 1) nd (6, 1) nd vertices (3, 1) nd (, 1). 30. An ellipse with foci (3, 0) nd (3, 6) nd vertices (3, 2) nd (3, 8). 31. A prbol with is the is nd pssing through the points (1, 0) nd (2, 4). 32. A prbol with verte t (6, 2), is the line 2, nd pssing through the point (0, 7). 33. An ellipse with vertices (1, 1), nd (, 1) tht psses through the origin. 34. A hperbol with vertices t (2, 3), nd (2, ) tht psses through the point (4, 0). C In Problems 3 40, find the coordintes of n foci reltive to the originl coordinte sstem: 3. Problem Problem Problem Problem Problem Problem 22 In Problems 41 44, use grphing utilit to find the coordintes of ll points of intersection to two deciml plces , , , , CHAPTER 9 GROUP ACTIVITY Focl Chords Mn of the pplictions of the conic sections re bsed on their reflective or focl properties. One of the interesting lgebric properties of the conic sections concerns their focl chords. If line through focus F contins two points G nd H of conic section, then the line segment GH is clled focl chord. Let G( 1, 1 ) nd H( 2, 2 ) be points on the grph of 2 4 such tht GH is focl chord. Let u denote the length of GF nd v the length of FH (see Fig. 1). FIGURE 1 Focl chord GH of the prbol 2 4. G F u v H (2, ) (A) Use the distnce formul to show tht u 1. (B) Show tht G nd H lie on the line m, where m ( 2 1 )/( 2 1 ). (C) Solve m for nd substitute in 2 4, obtining qudrtic eqution in. Eplin wh (D) Show tht. u 1 v 1 (u 2) 2 (E) Show tht u v 4. Eplin wh this implies tht u v 4, with equlit if nd onl u if u v 2. (F) Which focl chord is the shortest? Is there longest focl chord? 1 (G) Is constnt for focl chords of the ellipse? For focl chords of the hperbol? Obtin evidence u 1 v for our nswers b considering specific emples.