1 62 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Section 7-4 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 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 TRANSLATION FORMULAS 1. h 2. h k k It cn be shown tht these formuls hold for (h, k) locted nwhere in the originl coordinte sstem.
2 7-4 Trnsltion of Aes 63 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 1-1. FIGURE 2 ( 4) 2 ( 1) A(4, 1) 10 MATCHED 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
3 64 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY For prbols we hve 2 4 ( h) 2 4( k) 2 4 ( k) 2 4( h) For circles we hve 2 2 r 2 ( h) 2 ( k) 2 r 2 For ellipses we hve for b b b ( h) 2 ( k)2 ( h) 2 ( k)2 b 2 2 For hperbols we hve b b 2 1 ( h) 2 ( k)2 ( k) 2 ( h)2 Tble 1 summrizes these results with pproprite figures nd some properties discussed erlier. 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 2-3 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. Check b grphing on grphing utilit.
4 7-4 Trnsltion of Aes 6 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 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 Hperbols ( h) 2 ( k)2 ( k) 2 ( h)2 Center (h, k) Trnsverse is 2 Conjugte is 2b Center (h, k) Trnsverse is 2 Conjugte is 2b b (h, k) (h, k) b
5 66 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Solution Step 1. Complete the squre in eqution (2) reltive to ech vrible tht is squred in this cse : ( 3) 2 4( 2) Add 9 to both sides to complete the squre on the left side. (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): FIGURE Mking these substitutions in eqution (3) we obtin 2 4 (4) A( 2, 3) 0 the eqution of the prbol in the sstem. Step 3. Grph eqution (4) in the sstem following the process discussed in Section 7-1. The resulting grph is the grph of the originl eqution reltive to the originl coordinte sstem (Fig. 3). To check the grph in Figure 3 on grphing utilit, we cn solve either eqution (2) or eqution (3) for. Choosing eqution (2) hs the dded benefit of providing check of the derivtion of eqution (3). FIGURE ; Qudrtic eqution with 1, b 6, nd c (1) ( 4 1) 2(1) () 3 Figure 4 shows the grph of the two functions determined b eqution () nd the verte of the prbol.
6 7-4 Trnsltion of Aes 67 MATCHED 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. Check b grphing on grphing utilit. Solution Step 1. Complete the squre reltive to both nd ( 2 4 ) 4( 2 6 9) 36 9( 2 4 4) 4( 2 6 9) ( 2) 2 4( 3) 2 36 ( 2) 2 4 ( 3)2 9 1 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: FIGURE Mking these substitutions, we obtin F c 10 F c 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 7-3. The resulting grph is the grph of the originl eqution reltive to the originl coordinte sstem (Fig. ).
7 68 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY 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: c c 13 c 13 Thus, the coordintes in the trnslted sstem re F ( 13, 0) nd F( 13, 0) Now, use h 2 k 3 to obtin F ( 13 2, 3) nd F( 13 2, 3) FIGURE ; s the coordintes of the foci in the originl sstem. To check the grph in Figure, we return to the originl eqution nd use the qudrtic formul to solve for : ( ) 0 Write in the form 2 b c (4) ( ) The two functions determined b eqution (6) re grphed in Figure 6. (6) MATCHED 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. Eplore/Discuss 1 D If A 0 nd C 0, show tht the trnsltion of es, 2A E trnsforms the eqution A 2 C 2 D E F 0 2C into n eqution of the form A 2 C 2 K.
8 Finding Equtions of Conics 7-4 Trnsltion of Aes 69 We now reverse the problem: Given certin informtion bout conic in rectngulr coordinte sstem, find its eqution. EXAMPLE 4 Solution FIGURE 7 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. Locte the vertices, smptote rectngle, nd smptotes in the originl coordinte sstem [Fig. 7()], then sketch the hperbol nd trnslte the origin to the center of the hperbol [Fig. 7(b)]. 4 2 b 3 3 () Asmptote rectngle (b) Hperbol Net write the eqution of the hperbol in the trnslted sstem: 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), MATCHED 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.
9 70 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY Eplore/Discuss 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 interesting lines). (A) (B) (C) (D) (E) Answers to Mtched Problems ; prbol 2. ( 2) 2 4( 4); prbol ( 2, 4) ( 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 7-4 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) 9 16
10 7-4 Trnsltion of Aes ( 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 36, 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 (2, ), is the line 2, nd pssing through the point ( 2, 1). 26. A prbol with verte t (4, 1), is the line 1, nd pssing through the point (2, 3). 27. An ellipse with mjor is on the line 3, minor is on the line 2, length of mjor is 8, nd length of minor is An ellipse with mjor is on the line 4, minor is on the line 1, length of mjor is 4, nd length of minor is An ellipse with vertices (4, 7) nd (4, 3) nd foci (4, 6) nd (4, 2). 30. An ellipse with vertices ( 3, 1) nd (7, 1) nd foci ( 1, 1) nd (, 1). 31. A hperbol with trnsverse is on the line 2, length of trnsverse is 4, conjugte is on the line 3, nd length of conjugte is A hperbol with trnsverse is on the line, length of trnsverse is 6, conjugte is on the line 2, nd length of conjugte is An ellipse with the following grph: ( 3, 1) ( 2, 4) ( 2, 2) 34. An ellipse with the following grph: (, 2) ( 3, 1) ( 1, 1) ( 3, 3) ( 1, 2)
11 72 7 ADDITIONAL TOPICS IN ANALYTIC GEOMETRY 3. A hperbol with the following grph: ( 2, 4) (0, 2) 36. A hperbol with the following grph: (4, 4) (2, 2) C In Problems 37 42, find the coordintes of n foci reltive to the originl coordinte sstem. 37. Problem Problem Problem Problem Problem Problem 22 (2, 0) In Problems 43 46, find the coordintes of ll points of intersection to two deciml plces. (3, 1) (3, 3) (2, 2) , , , , Section 7- Prmetric Equtions Prmetric Equtions nd Plne Curves Projectile Motion FIGURE 1 Grph of t 1, t 2 2t, t. 10 Prmetric Equtions nd Plne Curves Consider the two equtions t 1 t 2 2t 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.
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