1 Appendi B. Conic Sections B B Conic Sections B. Conic Sections Recognize the four bsic conics: circles, prbols, ellipses, nd hperbols. Recognize, grph, nd write equtions of prbols (verte t origin). Recognize, grph, nd write equtions of ellipses (center t origin). Recognize, grph, nd write equtions of hperbols (center t origin). Introduction to Conic Sections Conic sections were discovered during the clssicl Greek period, which lsted from 600 to 00 B.C. B the beginning of the Alendrin period, enough ws known of conics for Apollonius (6 90 B.C.) to produce n eight-volume work on the subject. This erl Greek stud ws lrgel concerned with the geometric properties of conics. It ws not until the erl seventeenth centur tht the brod pplicbilit of conics becme pprent. A conic section (or simpl conic) cn be described s the intersection of plne nd double-npped cone. Notice from Figure B. tht in the formtion of the four bsic conics, the intersecting plne does not pss through the verte of the cone. When the plne does pss through the verte, the resulting figure is degenerte conic, s shown in Figure B.. Conic Sections FIGURE B. Degenerte Conics FIGURE B. There re severl ws to pproch the stud of conics. You could begin b defining conics in terms of the intersections of plnes nd cones, s the Greeks did, or ou could define them lgebricll, in terms of the generl second-degree eqution A B C D E F 0. However, ou will stud third pproch in which ech of the conics is defined s locus, or collection, of points stisfing certin geometric propert. For emple, in Section. ou sw how the definition of circle s the collection of ll points, tht re equidistnt from fied point h, k led esil to the stndrd eqution of circle, h k r. You will restrict our stud of conics in Appendi B. to prbols with vertices t the origin, nd ellipses nd hperbols with centers t the origin. In Appendi B., ou will look t the generl cses.
2 B Appendi B Conic Sections STUDY TIP Note tht the term prbol is technicl term used in mthemtics nd does not simpl refer to n U-shped curve. Prbols In Section., ou determined tht the grph of the qudrtic function given b f b c is prbol tht opens upwrd or downwrd. The definition of prbol given below is more generl in the sense tht it is independent of the orienttion of the prbol. Definition of Prbol A prbol is the set of ll points, in plne tht re equidistnt from fied line clled the directri nd fied point clled the focus (not on the line). The midpoint between the focus nd the directri is clled the verte, nd the line pssing through the focus nd the verte is clled the is of the prbol. Directri Focus Verte d Ais d d (, ) d Using this definition, ou cn derive the following stndrd form of the eqution of prbol. Stndrd Eqution of Prbol (Verte t Origin) The stndrd form of the eqution of prbol with verte t 0, 0 nd directri p is given b p, p 0. Verticl is For directri p, the eqution is given b p, p 0. Horizontl is The focus is on the is p units (directed distnce) from the verte. See Figure B.. = p, p 0 Focus (0, p) Verte (0, 0) (, ) p p Directri: = p Ais (, ) Directri: = p Verte (0, 0) p p Focus (p, 0) Ais = p, p 0 Prbol with Verticl Ais FIGURE B. Prbol with Horizontl Ais
3 Appendi B. Conic Sections B Emple Finding the Focus of Prbol Find the focus of the prbol whose eqution is. ( Focus 0, 8 = ( SOLUTION Becuse the squred term in the eqution involves, ou know tht the is is verticl, nd the eqution is of the form p. Stndrd form, verticl is You cn write the originl eqution in this form, s shown. Write originl eqution. Divide ech side b. 8. Write in stndrd form. FIGURE B. So, p 8. Becuse p is negtive, the prbol opens downwrd nd the focus of the prbol is 0, p 0, 8, s shown in Figure B.. Checkpoint Find the focus of the prbol whose eqution is. Verte = 8 Focus (, 0) (0, 0) FIGURE B.5 Emple Finding the Stndrd Eqution of Prbol Write the stndrd form of the eqution of the prbol with verte t the origin nd focus t, 0. SOLUTION The is of the prbol is horizontl, pssing through 0, 0 nd, 0, s shown in Figure B.5. So, the stndrd form is p. Stndrd form, horizontl is Becuse the focus is p units from the verte, the eqution is Stndrd form 8. Checkpoint Write the stndrd form of the eqution of the prbol with verte t the origin nd focus t 0,. Prbols occur in wide vriet of pplictions. For instnce, prbolic reflector cn be formed b revolving prbol bout its is. The resulting surfce hs the propert tht ll incoming rs prllel to the is re reflected through the focus of the prbol. This is the principle behind the construction of the prbolic mirrors used in reflecting telescopes. Conversel, the light rs emnting from the focus of the prbolic reflector used in flshlight re ll reflected prllel to one nother, s shown in Figure B.6. Focus Light source t focus FIGURE B.6 Ais Prbolic reflector: Light is reflected in prllel rs.
4 B Appendi B Conic Sections Ellipses Another bsic tpe of conic is clled n ellipse. Definition of n Ellipse An ellipse is the set of ll points, in plne the sum of whose distnces from two distinct fied points, clled foci, is constnt. (, ) Focus d d Focus Mjor is Verte Center Minor is Verte d + d is constnt. The line through the foci intersects the ellipse t two points, clled the vertices. The chord joining the vertices is clled the mjor is, nd its midpoint is clled the center of the ellipse. The chord perpendiculr to the mjor is t the center is clled the minor is of the ellipse. The endpoints of the minor is of n ellipse re commonl referred to s the co-vertices. You cn visulize the definition of n ellipse b imgining two thumbtcks plced t the foci, s shown in Figure B.7. If the ends of fied length of string re fstened to the thumbtcks nd the string is drwn tut with pencil, the pth trced b the pencil will be n ellipse. The stndrd form of the eqution of n ellipse tkes one of two forms, depending on whether the mjor is is horizontl or verticl. FIGURE B.7 Stndrd Eqution of n Ellipse (Center t Origin) The stndrd form of the eqution of n ellipse with the center t the origin nd mjor nd minor es of lengths nd b, respectivel where 0 < b <, is b or b. + b = (0, b) b + = (0, ) Verte (, 0) Verte ( c, 0) Verte (c, 0) (, 0) ( b, 0) (0, c) (0, c) (b, 0) (0, b ) Mjor is is horizontl. Minor is is verticl. (0, ) Verte Mjor is is verticl. Minor is is horizontl. The vertices nd foci lie on the mjor is, nd c units, respectivel, from the center. Moreover,, b, nd c re relted b the eqution c b.
5 Appendi B. Conic Sections B5 Emple Finding the Stndrd Eqution of n Ellipse (, 0) + b = (, 0) Find the stndrd form of the eqution of the ellipse tht hs mjor is of length 6 nd foci t, 0 nd, 0, s shown in Figure B.8. SOLUTION Becuse the foci occur t, 0 nd, 0, the center of the ellipse is 0, 0, nd the mjor is is horizontl. So, the ellipse hs n eqution of the form b. Stndrd form, horizontl mjor is Becuse the length of the mjor is is 6, ou hve 6, which implies tht. Moreover, the distnce from the center to either focus is c. Finll, ou hve b c 9 5. FIGURE B.8 Substituting nd ields the following eqution in stndrd form. b 5 5 Stndrd form This eqution simplifies to 9 5. Checkpoint Find the stndrd form of the eqution of the ellipse tht hs mjor is of length 8 nd foci t 0, nd 0,. Emple Sketching n Ellipse (0, 6) + 6 = (, 0) (, 0) 6 6 (0, 6) FIGURE B.9 Sketch the ellipse given b 6 nd identif the vertices. SOLUTION Begin b writing the eqution in stndrd form. Write originl eqution. Divide ech side b 6. Simplif. Write in stndrd form. Becuse the denomintor of the -term is greter thn the denomintor of the -term, ou cn conclude tht the mjor is is verticl. Also, becuse 6, the endpoints of the mjor is lie si units up nd down from the center 0, 0. So, the vertices of the ellipse re 0, 6 nd 0, 6. Similrl, becuse the denomintor of the -term is b, the endpoints of the minor is (or co-vertices) lie three units to the right nd left of the center t, 0 nd, 0. The ellipse is shown in Figure B.9. Checkpoint Sketch the ellipse given b 6, nd identif the vertices.
6 B6 Appendi B Conic Sections Hperbols The definition of hperbol is similr to tht of n ellipse. The distinction is tht, for n ellipse, the sum of the distnces between the foci nd point on the ellipse is constnt, wheres for hperbol, the difference of the distnces between the foci nd point on the hperbol is constnt. Definition of Hperbol A hperbol is the set of ll points, in plne the difference of whose distnces from two distinct fied points, clled foci, is constnt. Focus d (, ) d Focus Brnch Trnsverse is Brnch Verte Verte Center d d is constnt. c The grph of hperbol hs two disconnected prts, clled brnches. The line through the two foci intersects the hperbol t two points, clled vertices. The line segment connecting the vertices is clled the trnsverse is, nd the midpoint of the trnsverse is is clled the center of the hperbol. Stndrd Eqution of Hperbol (Center t Origin) The stndrd form of the eqution of hperbol with the center t the origin (where 0 nd b 0) is b or b. b = Verte: (, 0) Focus: ( c, 0) Trnsverse is Verte: (, 0) Focus: (c, 0) b = Verte: (0, ) Verte: (0, ) Focus: (0, c) Focus: (0, c) Trnsverse is The vertices nd foci re, respectivel, nd c units from the center. Moreover,, b, nd c re relted b the eqution b c.
7 Appendi B. Conic Sections B7 Emple 5 Finding the Stndrd Eqution of Hperbol (, 0) (, 0) FIGURE B.0 (, 0) (, 0) Find the stndrd form of the eqution of the hperbol with foci t, 0 nd, 0 nd vertices t, 0 nd, 0, s shown in Figure B.0. SOLUTION From the grph, c becuse the foci re three units from the center. Also, becuse the vertices re two units from the center. So, b c 9 5. Becuse the trnsverse is is horizontl, the stndrd form of the eqution is b. Stndrd form, horizontl trnsverse is Finll, substituting nd ou hve 5. b 5, Write in stndrd form. Checkpoint 5 Find the stndrd form of the eqution of the hperbol with foci t 0, nd 0, nd vertices t 0, nd 0,. An importnt id in sketching the grph of hperbol is the determintion of its smptotes, s shown in Figure B.. Ech hperbol hs two smptotes tht intersect t the center of the hperbol. Furthermore, the smptotes pss through the corners of rectngle of dimensions b b. The line segment of length b, joining 0, b nd 0, b or b, 0 nd b, 0, is referred to s the conjugte is of the hperbol. b = Asmptote: b = b = (, 0) (0, b) (0, b) (, 0) Asmptote: b = ( b, 0) (0, ) (0, ) Asmptote: = b (b, 0) Asmptote: = b Trnsverse is is horizontl. FIGURE B. Trnsverse is is verticl. Asmptotes of Hperbol (Center t Origin) b nd b Trnsverse is is horizontl. b nd b Trnsverse is is verticl.
8 B8 Appendi B Conic Sections Emple 6 Sketching Hperbol Sketch the hperbol whose eqution is 6. SOLUTION Write originl eqution. Divide ech side b 6. Write in stndrd form. Becuse the -term is positive, ou cn conclude tht the trnsverse is is horizontl nd the vertices occur t, 0 nd, 0. Moreover, the endpoints of the conjugte is occur t 0, nd 0,, nd ou cn sketch the rectngle shown in Figure B.. Finll, b drwing the smptotes through the corners of this rectngle, ou cn complete the sketch shown in Figure B.. 6 (0, ) (, 0) (, 0) = 6 6 (0, ) 6 FIGURE B. FIGURE B. Checkpoint 6 Sketch the hperbol whose eqution is 9 9. Emple 7 Finding the Stndrd Eqution of Hperbol Find the stndrd form of the eqution of the hperbol tht hs vertices t 0, nd 0, nd smptotes nd, s shown in Figure B.. = (0, ) SOLUTION Becuse the trnsverse is is verticl, the smptotes re of the form = FIGURE B. (0, ) b nd Trnsverse is is verticl. Using the fct tht nd, ou cn determine tht b. Becuse, ou cn determine tht b. Finll, ou cn conclude tht the hperbol hs the eqution. b. Write in stndrd form. Checkpoint 7 Find the stndrd form of the eqution of the hperbol tht hs vertices t 5, 0 nd 5, 0 nd smptotes nd.
9 Appendi B. Conic Sections B9 SKILLS WARM UP B. The following wrm-up eercises involve skills tht were covered in erlier sections. You will use these skills in the eercise set for this section. For dditionl help, review Sections 0.7,., nd.. In Eercises, rewrite the eqution so tht it hs no frctions In Eercises 5 8, solve for c. (Assume c > 0. ) 5. c 6. c 7. c 8. c In Eercises 9 nd 0, find the distnce between the point nd the origin. 9. 0, 0., 0 Eercises B. Mtching In Eercises 8, mtch the eqution with its grph. [The grphs re lbeled (), (b), (c), (d), (e), (f), (g), nd (h).] () (c) (e) 6 6 (b) (d) (f) Finding the Verte nd Focus of Prbol In Eercises 9 6, find the verte nd focus of the prbol nd sketch its grph. See Emple (g) (h) 8 Finding the Stndrd Eqution of Prbol In Eercises 7 6, find the stndrd form of the eqution of the prbol with the given chrcteristic(s) nd verte t the origin. See Emple. 7. Focus: 8. Focus: 0, 0,
10 B0 Appendi B Conic Sections 9. Focus:, 0 0. Focus:. Directri:. Directri:. Directri:. Directri: 5. Psses through the point, 6; horizontl is 6. Psses through the point, ; verticl is Finding the Stndrd Eqution of n Ellipse In Eercises 7, find the stndrd form of the eqution of the ellipse with the given chrcteristics nd center t the origin. See Emple. 7. Vertices: 0, ±; minor is of length 8. Vertices: ±, 0; minor is of length 9. Vertices: ±5, 0; foci: ±, 0 0. Vertices: 0, ±0; foci: 0, ±. Foci: ±5, 0; mjor is of length. Foci: ±, 0; mjor is of length 8. Vertices: 0, ±5; psses through the point,. Mjor is verticl; psses through the points 0, nd, 0 Sketching n Ellipse In Eercises 5, find the center nd vertices of the ellipse nd sketch its grph. See Emple Finding the Stndrd Eqution of Hperbol In Eercises 6, find the stndrd form of the eqution of the hperbol with the given chrcteristics nd center t the origin. See Emple 5.. Vertices: 0, ±; foci: 0, ±. Vertices: 0, ±5; foci: 0, ±8 5. Vertices: ±6, 0; foci: ±9, 0 6. Vertices: ±, 0; foci: ±5, 0 Sketching Hperbol In Eercises 7 5, find the center nd vertices of the hperbol nd sketch its grph. See Emple , Finding the Stndrd Eqution of Hperbol In Eercises 55 60, find the stndrd form of the eqution of the hperbol with the given chrcteristics nd center t the origin. See Emple Vertices: ±, 0; smptotes: ± 56. Vertices: 0, ±; smptotes: ± 57. Foci: 0, ±; smptotes: ± 58. Foci: ±0, 0; smptotes: ± 59. Vertices: 0, ±; psses through the point, Vertices: ±, 0; psses through the point, 6. Stellite Antenn The receiver in prbolic television dish ntenn is feet from the verte nd is locted t the focus (see figure). Write n eqution for cross section of the reflector. (Assume tht the dish is directed upwrd nd the verte is t the origin.) 6. Suspension Bridge Ech cble of the Golden Gte Bridge is suspended (in the shpe of prbol) between two towers tht re 80 meters prt. The top of ech tower is 5 meters bove the rodw (see figure). The cbles touch the rodw t the midpoint between the towers. Write n eqution for the prbolic shpe of ech cble. ( 60, 5) Receiver Cble 6. Architecture A fireplce rch is to be constructed in the shpe of semiellipse. The opening is to hve height of feet t the center nd width of 5 feet long the bse (see figure). The contrctor drws the outline of the ellipse b the method shown in Figure B.7. Where should the tcks be plced nd wht should be the length of the piece of string? Rodw (60, 5) ft
11 Appendi B. Conic Sections B 6. Mountin Tunnel A semiellipticl rch over tunnel for rod through mountin hs mjor is of 00 feet, nd its height t the center is 0 feet (see figure). Determine the height of the rch 5 feet from the edge of the tunnel. Using Lter Rect In Eercises 67 70, sketch the ellipse using the lter rect (see Eercise 66) Think About It Consider the ellipse 0 ft 5 ft 00 ft 65. Sketching n Ellipse Sketch grph of the ellipse tht consists of ll points, such tht the sum of the distnces between, nd two fied points is 5 units nd the foci re locted t the centers of the two sets of concentric circles, s shown in the figure Is this ellipse better described s elongted or nerl circulr? Eplin our resoning. 7. Nvigtion Long-rnge nvigtion for ircrft nd ships is ccomplished b snchronized pulses trnsmitted b widel seprted trnsmitting sttions. These pulses trvel t the speed of light (86,000 miles per second). The difference in the times of rrivl of these pulses t n ircrft or ship is constnt on hperbol hving the trnsmitting sttions s foci. Assume tht two sttions 00 miles prt re positioned on rectngulr coordinte sstem t points with coordintes 50, 0 nd 50, 0 nd tht ship is trveling on pth with coordintes, 75 (see figure). Find the -coordinte of the position of the ship if the time difference between the pulses from the trnsmitting sttions is 000 microseconds (0.00 second) Think About It A line segment through focus of n ellipse with endpoints on the ellipse nd perpendiculr to its mjor is is clled ltus rectum of the ellipse. An ellipse hs two lter rect. Knowing the length of the lter rect is helpful in sketching n ellipse becuse this informtion ields other points on the curve (see figure). Show tht the length of ech ltus rectum is b. Lter rect Hperbolic Mirror A hperbolic mirror (used in some telescopes) hs the propert tht light r directed t one focus will be reflected to the other focus (see figure). The focus of the hperbolic mirror hs coordintes, 0. Find the verte of the mirror if its mount t the top edge of the mirror hs coordintes, F F (, ) (, 0) (, 0) 7. Writing Eplin how the centrl rectngle of hperbol cn be used to sketch its smptotes.
12 B Appendi B Conic Sections B. Conic Sections nd Trnsltions Recognize equtions of conics tht hve been shifted verticll or horizontll in the plne. Write nd grph equtions of conics tht hve been shifted verticll or horizontll in the plne. Verticl nd Horizontl Shifts of Conics In Appendi B., ou studied conic sections whose grphs were in stndrd position. In this section, ou will stud the equtions of conic sections tht hve been shifted verticll or horizontll in the plne. The following summr lists the stndrd forms of the equtions of the four bsic conics. ( h) + ( k) = r Stndrd Forms of Equtions of Conics Circle: Center h, k; Rdius r; See Figure B.5. Prbol: Verte h, k; Directed distnce from verte to focus p (h, k) r ( h) = p( k) p > 0 ( k) = p( h) FIGURE B.5 p > 0 Verte: (h, k) Focus: (h, k + p) Verte: (h, k) Focus: (h + p, k) Ellipse: Center h, k; Mjor is length ; Minor is length b ( h) ( k) + b = ( h) ( k) b + = (h, k) b (h, k) b Hperbol: Center h, k; Trnsverse is length ; Conjugte is length b ( h) ( k) b = ( k) ( h) b = (h, k) b (h, k) b
13 Appendi B. Conic Sections nd Trnsltions B Emple Equtions of Conic Sections 5 (, ) 5 6 FIGURE B.6 FIGURE B.7 ( ) + ( + ) = ( ) = ( )( ) (, ) (, ) 5 p = Describe the trnsltion of the grph of ech conic.. b. c. d. SOLUTION. The grph of is circle whose center is the point, nd whose rdius is, s shown in Figure B.6. The grph of the circle hs been shifted one unit to the right nd two units downwrd from stndrd position. b. The grph of is prbol whose verte is the point,. The is of the prbol is verticl. Moreover, becuse p, it follows tht the focus lies below the verte, s shown in Figure B.7. The grph of the prbol hs been reflected in the -is, nd shifted two units to the right nd three units upwrd from stndrd position. c. The grph of is hperbol whose center is the point,. The trnsverse is is horizontl with length of. The conjugte is is verticl with length of 6, s shown in Figure B.8. The grph of the hperbol hs been shifted three units to the right nd two units upwrd from stndrd position. d. The grph of is n ellipse whose center is the point, ). The mjor is of the ellipse is horizontl with length of 6. The minor is of the ellipse is verticl with length of, s shown in Figure B.9. The grph of the ellipse hs been shifted two units to the right nd one unit upwrd from stndrd position. 8 6 ( ) ( ) = (, ) ( ) ( ) + = (, ) 5 FIGURE B.8 FIGURE B.9 Checkpoint Describe the trnsltion of the grph of 5.
14 B Appendi B Conic Sections Writing Equtions of Conics in Stndrd Form Emple Finding the Stndrd Form of Prbol (, ) (, 0) ( ) = ( )( ) FIGURE B.0 Find the verte nd focus of the prbol given b 0. SOLUTION Complete the squre to write the eqution in stndrd form. 0 Write originl eqution. Group terms. Complete the squre. Write in completed squre form. Stndrd form, h p k From this stndrd form, it follows tht h, k, nd p. Becuse the is is verticl nd p is negtive, the prbol opens downwrd. The verte is h, k, nd the focus is h, k p, 0. (See Figure B.0.) Checkpoint Find the verte nd focus of the prbol given b 5 0. In Emples (b) nd, p is the directed distnce from the verte to the focus. Becuse the is of the prbol is verticl nd p, the focus is one unit below the verte, nd the prbol opens downwrd. Emple Sketching n Ellipse ( 5, ) 5 ( + ) ( ) + = (, ) (, ) (, 0) FIGURE B. (, ) Sketch the ellipse given b SOLUTION Complete the squre to write the eqution in stndrd form Write originl eqution. Group terms. Fctor out of -terms. Complete the squres. From this stndrd form, it follows tht the center is h, k,. Becuse the denomintor of the -term is, the endpoints of the mjor is lie two units to the right nd left of the center. Similrl, becuse the denomintor of the -term is b, the endpoints of the minor is lie one unit up nd down from the center. The ellipse is shown in Figure B.. Checkpoint Write in completed squre form. Divide ech side b. h k b Sketch the ellipse given b
15 Appendi B. Conic Sections nd Trnsltions B5 Emple Sketching Hperbol 6 FIGURE B. ( + ) ( ) (/) = (, ) 6 (, 5) (, ) Sketch the hperbol given b 0. SOLUTION Complete the squre to write the eqution in stndrd form Write originl eqution. Group terms. Fctor out of -terms. Complete the squres. Divide ech side b 9. Chnge to. k From the stndrd form, it follows tht the trnsverse is is verticl nd the center lies t h, k,. Becuse the denomintor of the -term is, ou know tht the vertices lie three units bove nd below the center. Vertices:, nd, 5 To sketch the hperbol, drw rectngle whose top nd bottom pss through the vertices. Becuse the denomintor of the -term is b, locte the sides of the rectngle units to the right nd left of the center, s shown in Figure B.. Finll, sketch the smptotes b drwing lines through the opposite corners of the rectngle. Using these smptotes, ou cn complete the grph of the hperbol, s shown in Figure B.. Write in completed squre form. h b Checkpoint Sketch the hperbol given b 6 0. To find the foci in Emple, first find b c. So, c b c. Recll from Appendi B. tht c 9 9 c 5 c 5. Becuse the trnsverse is is verticl, the foci lie c units bove nd below the center. 5 5 Foci:, nd,
16 B6 Appendi B Conic Sections Emple 5 Writing the Eqution of n Ellipse (, ) 5 (, ) FIGURE B. Write the stndrd form of the eqution of the ellipse whose vertices re, nd,. The length of the minor is of the ellipse is, s shown in Figure B.. SOLUTION center is The center of the ellipse lies t the midpoint of its vertices. So, the h, k, Center Becuse the vertices lie on verticl line nd re si units prt, it follows tht the mjor is is verticl nd hs length of 6. So,. Moreover, becuse the minor is hs length of, it follows tht b, which implies tht b. So, ou cn conclude tht the stndrd form of the eqution of the ellipse is h b,. k. Mjor is is verticl. Write in stndrd form. Hperbolic orbit Verte Ellipticl orbit p Sun (Focus) Prbolic orbit FIGURE B. Checkpoint 5 Write the stndrd form of the eqution of the ellipse whose vertices re, nd,. The length of the minor is of the ellipse is. An interesting ppliction of conic sections involves the orbits of comets in our solr sstem. Of the 60 comets identified prior to 970, 5 hve ellipticl orbits, 95 hve prbolic orbits, nd 70 hve hperbolic orbits. For emple, Hlle s comet hs n ellipticl orbit, nd reppernce of this comet cn be predicted ever 76 ers. The center of the sun is focus of ech of these orbits, nd ech orbit hs verte t the point where the comet is closest to the sun, s shown in Figure B.. If p is the distnce between the verte nd the focus (in meters), nd v is the speed of the comet t the verte (in meters per second), then the tpe of orbit is determined s follows.. Ellipse:. Prbol: v < GM p v GM p. Hperbol: v > GM p In these epressions, M kilogrms (the mss of the sun) nd G cubic meter per kilogrm-second squred (the universl grvittionl constnt).
17 Appendi B. Conic Sections nd Trnsltions B7 SKILLS WARM UP B. The following wrm-up eercises involve skills tht were covered in erlier sections. You will use these skills in the eercise set for this section. For dditionl help, review Section B.. In Eercises 0, identif the conic represented b the eqution Eercises B. Equtions of Conic Sections In Eercises 6, describe the trnsltion of the grph of the conic from the stndrd position. See Emple... ( + ) + ( ) = ( + ) ( ) = ( ) ( + ) + = ( ) = ()( + ) 6 ( ) ( + ) + = ( + ) ( + ) = 9 Finding the Stndrd Form of Prbol In Eercises 7 6, find the verte, focus, nd directri of the prbol. Then sketch its grph. See Emple Finding the Stndrd Form of Prbol In Eercises 7, find the stndrd form of the eqution of the prbol with the given chrcteristics. 7. Verte:, ; focus:, 8. Verte:, ; focus:, 0 9. Verte: 0, ; directri: 0. Verte:, ; directri:. Focus:, ; directri:. Focus: 0, 0; directri:. Verte: 0, ; psses through, 0 nd, 0. Verte:, ; psses through 0, 0 nd, 0
18 B8 Appendi B Conic Sections Sketching n Ellipse In Eercises 5, find the center, foci, nd vertices of the ellipse. Then sketch its grph. See Emple Sketching Hperbol In Eercises, find the center, vertices, nd foci of the hperbol. Then sketch its grph, using smptotes s sketching ids. See Emple Finding the Stndrd Eqution of n Ellipse In Eercises 5, find the stndrd form of the eqution of the ellipse with the given chrcteristics. See Emple 5.. Vertices: 0,,, ; minor is of length. Vertices:,,, 9; minor is of length 6 5. Foci: 0, 0,, 0; mjor is of length 8 6. Foci: 0, 0, 0, 8; mjor is of length 6 7. Center:, ; verte:, ; minor is of length 8. Center:, 0; verte:, ; minor is of length 6 9. Center:, ; c; foci:,, 5, 50. Center: 0, ; c; vertices,,, 5. Vertices: 5, 0, 5, ; endpoints of minor is: 0, 6, 0, 6 5. Vertices:, 0, 0, 0; endpoints of minor is: 6,, 6, Finding the Stndrd Eqution of Hperbol In Eercises 5 60, find the stndrd form of the eqution of the hperbol with the given chrcteristics. 5. Vertices:, 0, 6, 0; foci: 0, 0, 8, 0 5. Vertices:,,, ; foci:, 5,, Vertices:,,, 9; foci:, 0,, Vertices:,,, ; foci:,,, 57. Vertices:,,, ; psses through 0, Vertices:,,, ; psses through, 59. Vertices: 0,, 6, ; smptotes:, 60. Vertices:, 0,, ; smptotes:, Identifing Conics In Eercises 6 68, identif the conic b writing the eqution in stndrd form. Then sketch its grph Stellite Orbit A stellite in 00-mile-high circulr orbit round Erth hs velocit of pproimtel 7,500 miles per hour. When this velocit is multiplied b, the stellite hs the minimum velocit necessr to escpe Erth s grvit, nd it follows prbolic pth with the center of Erth s the focus (see figure). Circulr orbit 00 Prbolic pth Not drwn to scle () Find the escpe velocit of the stellite. (b) Find n eqution of its pth (ssume the rdius of Erth is 000 miles).
19 Appendi B. Conic Sections nd Trnsltions B9 70. Fluid Flow Wter is flowing from horizontl pipe 8 feet bove the ground. The flling strem of wter hs the shpe of prbol whose verte 0, 8 is t the end of the pipe (see figure). The strem of wter strikes the ground t the point 0, 0. Find the eqution of the pth tken b the wter Eccentricit of n Ellipse In Eercises 7 78, the fltness of n ellipse is mesured b its eccentricit e, defined b e c where 0 < e <., 8 ft Plnetr Motion The dwrf plnet Pluto moves in n ellipticl orbit with the sun t one of the foci (see figure). The length of hlf of the mjor is is miles nd the eccentricit is 0.9. Find the shortest distnce nd the gretest distnce between Pluto nd the sun. 75. Plnetr Motion Sturn moves in n ellipticl orbit with the sun t one of the foci (see figure). The shortest distnce nd the gretest distnce between Sturn nd the sun re kilometers nd kilometers, respectivel. Find the eccentricit of the orbit. 76. Stellite Orbit The first rtificil stellite to orbit Erth ws Sputnik I (lunched b the former Soviet Union in 957). Its highest point bove Erth s surfce ws 588 miles, nd its lowest point ws miles (see figure). Assume tht the center of Erth is one of the foci of the ellipticl orbit nd tht the rdius of Erth is 000 miles. Find the eccentricit of the orbit. When n ellipse is nerl circulr, e is close to 0. When n ellipse is elongted, e is close to (see figures). Focus c c e = : close to 0 c c e = : close to 7. Find n eqution of the ellipse with vertices ±5, 0 nd eccentricit e Find n eqution of the ellipse with vertices 0, ±8 nd eccentricit e. 7. Plnetr Motion Erth moves in n ellipticl orbit with the sun t one of the foci (see figure). The length of hlf of the mjor is is miles nd the eccentricit is Find the shortest distnce (perihelion) nd the gretest distnce (phelion) between Erth nd the sun. Figure for 7 75 Sun Not drwn to scle 77. Alternte Form of Eqution of n Ellipse Show tht the eqution of n ellipse cn be written s h miles Not drwn to scle 588 miles k e. 78. Comet Orbit Hlle s comet hs n ellipticl orbit with the sun t one focus. The eccentricit of the orbit is pproimtel The length of the mjor is of the orbit is pproimtel 5.88 stronomicl units. (An stronomicl unit is bout 9 million miles.) Find the stndrd form of the eqution of the orbit. Plce the center of the orbit t the origin nd plce the mjor is on the -is. 79. Austrlin Footbll In Austrli, footbll b Austrlin Rules (or rugb) is pled on ellipticl fields. The fields cn be mimum of 70 rds wide nd mimum of 00 rds long. Let the center of field of mimum size be represented b the point 0, 85. Find the stndrd form of the eqution of the ellipse tht represents this field. (Source: Austrlin Footbll Legue)
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
Lerning Objectives Loci nd Conics Lesson 3: The Ellipse Level: Preclculus Time required: 120 minutes In this lesson, students will generlize their knowledge of the circle to the ellipse. The prmetric nd
PROBLEMS - APPLICATIONS OF DERIVATIVES Pge ( ) Wter seeps out of conicl filter t the constnt rte of 5 cc / sec. When the height of wter level in the cone is 5 cm, find the rte t which the height decreses.
38 Chpter P Prerequisites P.4 Opertions with Polynomils Wht you should lern: Write polynomils in stndrd form nd identify the leding coefficients nd degrees of polynomils Add nd subtrct polynomils Multiply
0CH_PHClter_TMSETE_ 3//00 :3 PM Pge Grphs on Logrithmic nd Semilogrithmic Pper OBJECTIVES When ou hve completed this chpter, ou should be ble to: Mke grphs on logrithmic nd semilogrithmic pper. Grph empiricl
LECTURE #05 Chpter 3: Lttice Positions, Directions nd Plnes Lerning Objective To describe the geometr in nd round unit cell in terms of directions nd plnes. 1 Relevnt Reding for this Lecture... Pges 64-83.
10.6 Applictions of Qudrtic Equtions In this section we wnt to look t the pplictions tht qudrtic equtions nd functions hve in the rel world. There re severl stndrd types: problems where the formul is given,
33337_0P03.qp 2/27/06 24 9:3 AM Chpter P Pge 24 Prerequisites P.3 Polynomils nd Fctoring Wht you should lern Polynomils An lgeric epression is collection of vriles nd rel numers. The most common type of
Nme Chpter Eponentil nd Logrithmic Functions Section. Eponentil Functions nd Their Grphs Objective: In this lesson ou lerned how to recognize, evlute, nd grph eponentil functions. Importnt Vocbulr Define
5.2. LINE INTEGRALS 265 5.2 Line Integrls 5.2.1 Introduction Let us quickly review the kind of integrls we hve studied so fr before we introduce new one. 1. Definite integrl. Given continuous rel-vlued
5- Trigonometric Functions Section 5- Trigonometric Functions Definition of the Trigonometric Functions Clcultor Evlution of Trigonometric Functions Definition of the Trigonometric Functions Alternte Form
mth ppliction: volumes of revolution, prt ii Volumes of Revolution: The Disk Method One of the simplest pplictions of integrtion (Theorem ) nd the ccumultion process is to determine so-clled volumes of
MAT 0630 INTERNET RESOURCES, REVIEW OF CONCEPTS AND COMMON MISTAKES PROF. BOYAN KOSTADINOV NEW YORK CITY COLLEGE OF TECHNOLOGY, CUNY Contents 1. ACT Compss Prctice Tests 1 2. Common Mistkes 2 3. Distributive
Review Problems for the Finl of Mth, Fll The following is collection of vrious types of smple problems covering sections.,.5, nd.7 6.6 of the text which constitute only prt of the common Mth Finl. Since
2010 F = m Exm 1 AAPT UNITED STATES PHYSICS TEAM AIP 2010 Enti non multiplicnd sunt preter necessittem 2010 F = m Contest 25 QUESTIONS - 75 MINUTES INSTRUCTIONS DO NOT OPEN THIS TEST UNTIL YOU ARE TOLD
Mth 4, Homework Assignment. Prove tht two nonverticl lines re perpendiculr if nd only if the product of their slopes is. Proof. Let l nd l e nonverticl lines in R of slopes m nd m, respectively. Suppose
698 (3 0) Chapter 3 Nonlinear Sstems and the Conic Sections 49. Fencing a rectangle. If 34 ft of fencing are used to enclose a rectangular area of 72 ft 2, then what are the dimensions of the area? 50.
Appendi D: Completing the Squre nd the Qudrtic Formul Fctoring qudrtic epressions such s: + 6 + 8 ws one of the topics introduced in Appendi C. Fctoring qudrtic epressions is useful skill tht cn help you
Mth 135 Circles nd Completing the Squre Exmples A perfect squre is number such tht = b 2 for some rel number b. Some exmples of perfect squres re 4 = 2 2, 16 = 4 2, 169 = 13 2. We wish to hve method for
1 Exmple A rectngulr box without lid is to be mde from squre crdbord of sides 18 cm by cutting equl squres from ech corner nd then folding up the sides. 1 Exmple A rectngulr box without lid is to be mde
Algebr Review How well do you remember your lgebr? 1 The Order of Opertions Wht do we men when we write + 4? If we multiply we get 6 nd dding 4 gives 10. But, if we dd + 4 = 7 first, then multiply by then
www.mthsbo.org.uk CORE SUMMARY NOTES Functions A function is rule which genertes ectl ONE OUTPUT for EVERY INPUT. To be defined full the function hs RULE tells ou how to clculte the output from the input
Fctoring Polynomils Some definitions (not necessrily ll for secondry school mthemtics): A polynomil is the sum of one or more terms, in which ech term consists of product of constnt nd one or more vribles
Physics 43 Homework Set 9 Chpter 4 Key. The wve function for n electron tht is confined to x nm is. Find the normliztion constnt. b. Wht is the probbility of finding the electron in. nm-wide region t x
Propgtion of Uncertint through Mthemticl Opertions Since the untit of interest in n eperiment is rrel otined mesuring tht untit directl, we must understnd how error propgtes when mthemticl opertions re
9 CONTINUOUS DISTIBUTIONS A rndom vrible whose vlue my fll nywhere in rnge of vlues is continuous rndom vrible nd will be ssocited with some continuous distribution. Continuous distributions re to discrete
Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 9 The Force Method of Anlysis: Bems (Continued) Version CE IIT, Khrgpur Instructionl Objectives
Review guide for the finl exm in Mth 33 1 Bsic mteril. This review includes the reminder of the mteril for mth 33. The finl exm will be cumultive exm with mny of the problems coming from the mteril covered
Conic Sections. Distance Formula and Circles. More on the Parabola. The Ellipse and Hperbola. Nonlinear Sstems of Equations in Two Variables. Nonlinear Inequalities and Sstems of Inequalities In Chapter,
Bsic Anlysis of Autrky nd Free Trde Models AUTARKY Autrky condition in prticulr commodity mrket refers to sitution in which country does not engge in ny trde in tht commodity with other countries. Consequently
Order of Opertions r of Opertions Alger P lese Prenthesis - Do ll grouped opertions first. E cuse Eponents - Second M D er Multipliction nd Division - Left to Right. A unt S hniqu Addition nd Sutrction
PHY 222 Lb 8 MOTION OF ELECTRONS IN ELECTRIC AND MAGNETIC FIELDS Nme: Prtners: INTRODUCTION Before coming to lb, plese red this pcket nd do the prelb on pge 13 of this hndout. From previous experiments,
Volumes s integrls of cross-sections (ect. 6.1) Te volume of simple regions in spce Volumes integrting cross-sections: Te generl cse. Certin regions wit oles. Volumes s integrls of cross-sections (ect.
dius of the Erth - dii Used in Geodesy Jmes. Clynch Februry 006 I. Erth dii Uses There is only one rdius of sphere. The erth is pproximtely sphere nd therefore, for some cses, this pproximtion is dequte.
The Velocity Fctor of n Insulted Two-Wire Trnsmission Line Problem Kirk T. McDonld Joseph Henry Lbortories, Princeton University, Princeton, NJ 08544 Mrch 7, 008 Estimte the velocity fctor F = v/c nd the
Phsics 927 Section 1: Crstl Structure A solid is sid to be crstl if toms re rrnged in such w tht their positions re ectl periodic. This concept is illustrted in Fig.1 using two-dimensionl (2D) structure.
EQUATIONS OF LINES AND PLANES MATH 195, SECTION 59 (VIPUL NAIK) Corresponding mteril in the ook: Section 12.5. Wht students should definitely get: Prmetric eqution of line given in point-direction nd twopoint
Vectors mesurement which onl descries the mgnitude (i.e. size) of the oject is clled sclr quntit, e.g. Glsgow is 11 miles from irdrie. vector is quntit with mgnitude nd direction, e.g. Glsgow is 11 miles
Lecture 5 Inner Product Let us strt with the following problem. Given point P R nd line L R, how cn we find the point on the line closest to P? Answer: Drw line segment from P meeting the line in right
REVIEW OF CONIC SECTIONS In this section we give geometric definitions of parabolas, ellipses, and hperbolas and derive their standard equations. The are called conic sections, or conics, because the result
LECTURE COMPONENTS: COMBINED LOADING Third Edition A. J. Clrk School of Engineering Deprtment of Civil nd Environmentl Engineering 24 Chpter 8.4 by Dr. Ibrhim A. Asskkf SPRING 2003 ENES 220 Mechnics of
Summer Assignment Wrm-up for Differentil Clculus Who should complete this pcket? Students who hve completed Functions or Honors Functions nd will be tking Differentil Clculus in the fll of 015. Due Dte:
Physics 10 Jonthn Dowling Physics 10 Lecture Electric Fields Chrles-Augustin de Coulomb (1736-1806) Jnury 17, 07 Version: 1/17/07 Wht re we going to lern? A rod mp Electric chrge Electric force on other
2 Rtionl Numbers Integers such s 5 were importnt when solving the eqution x+5 = 0. In similr wy, frctions re importnt for solving equtions like 2x = 1. Wht bout equtions like 2x + 1 = 0? Equtions of this
PROBLEMS 0 ELLIPSE Pge 1 ( 1 ) The edpoits A d B of AB re o the X d Yis respectivel If AB > 0 > 0 d P divides AB from A i the rtio : the show tht P lies o the ellipse 1 ( ) If the feet of the perpediculrs
Vectors 2. Recp of vectors Vectors re directed line segments - they cn be represented in component form or by direction nd mgnitude. We cn use trigonometry nd Pythgors theorem to switch between the forms
A P P E N D I X A Vectors CONTENTS A.1 Scling vector................................................ 321 A.2 Unit or Direction vectors...................................... 321 A.3 Vector ddition.................................................
Brillouin Zones Physics 3P41 Chris Wiebe Direct spce to reciprocl spce * = 2 i j πδ ij Rel (direct) spce Reciprocl spce Note: The rel spce nd reciprocl spce vectors re not necessrily in the sme direction
Circuit nd Electromgnetic System Design Notes Note 61 3 July 009 Incresing Q of Wveguide Pulse-Compression Cvities Crl E. Bum University of New Mexico Deprtment of Electricl nd Computer Engineering Albuquerque
Rotation of Aes ROTATION OF AES Rotation of Aes For a discussion of conic sections, see Calculus, Fourth Edition, Section 11.6 Calculus, Earl Transcendentals, Fourth Edition, Section 1.6 In precalculus
Obits nd Keple s Lws This web pge intoduces some of the bsic ides of obitl dynmics. It stts by descibing the bsic foce due to gvity, then consides the ntue nd shpe of obits. The next section consides how
Binry Representtion of Numbers Autr Kw After reding this chpter, you should be ble to: 1. convert bse- rel number to its binry representtion,. convert binry number to n equivlent bse- number. In everydy
From the Mthemtics Techer, Vol. 95, No. 8, pges 568-574 Wlter Dodge (not pictured) nd Steve Viktor Thinking out of the Bo... Problem It s richer problem thn we ever imgined The bo problem hs been stndrd
Version 001 Summer Reiew #03 tubmn (IBII20142015) 1 This print-out should he 35 questions. Multiple-choice questions my continue on the next column or pge find ll choices before nswering. Concept 20 P03
314 CHAPTER 4 Vector Spces 9. A mtrix of the form 0 0 b c 0 d 0 0 e 0 f g 0 h 0 cnnot be invertible. 10. A mtrix of the form bc d e f ghi such tht e bd = 0 cnnot be invertible. 4.11 Inner Product Spces
Version PEVIEW Prctice 7 crroll (08) his print-out should he 5 questions. Multiple-choice questions y continue on the next colun or pge find ll choices before nswering. Atwood Mchine 05 00 0.0 points A
7 CHAPTER THREE. Cross Product Given two vectors = (,, nd = (,, in R, the cross product of nd written! is defined to e: " = (!,!,! Note! clled cross is VECTOR (unlike which is sclr. Exmple (,, " (4,5,6
Pentominoes Bruce Bguley Cscde Mth Systems, LLC Astrct. Pentominoes nd their reltives the polyominoes, polycues, nd polyhypercues will e used to explore nd pply vrious importnt mthemticl concepts. In this
7. Parabolas 0 7. Parabolas We have alread learned that the graph of a quadratic function f() = a + b + c (a 0) is called a parabola. To our surprise and delight, we ma also define parabolas in terms of
Helicopter Theme nd Vritions Or, Some Experimentl Designs Employing Pper Helicopters Some possible explntory vribles re: Who drops the helicopter The length of the rotor bldes The height from which the
smi9885_ch09b.qd 5/7/0 :5 PM Page 760 760 Chapter 9 Parametric Equations and Polar Coordinates 9.5 CALCULUS AND POLAR COORDINATES Now that we have introduced ou to polar coordinates and looked at a variet
Distributions Recll tht n integrble function f : R [,] such tht R f()d = is clled probbility density function (pdf). The distribution function for the pdf is given by F() = (corresponding to the cumultive
HPTER 9: Moments of nerti! Moment of nerti of res! Second Moment, or Moment of nerti, of n re! Prllel-is Theorem! Rdius of Grtion of n re! Determintion of the Moment of nerti of n re ntegrtion! Moments
The Sclr Product 9.3 Introduction There re two kinds of multipliction involving vectors. The first is known s the sclr product or dot product. This is so-clled becuse when the sclr product of two vectors
Drwing Digrms From Lbelled Grphs Jérôme Thièvre 1 INA, 4, venue de l Europe, 94366 BRY SUR MARNE FRANCE Anne Verroust-Blondet 2 INRIA Rocquencourt, B.P. 105, 78153 LE CHESNAY Cedex FRANCE Mrie-Luce Viud
10 AREA AND VOLUME 1 The Tower of Pis is circulr bell tower. Construction begn in the 1170s, nd the tower strted lening lmost immeditely becuse of poor foundtion nd loose soil. It is 56.7 metres tll, with
APPLICATION OF INTEGRALS 59 Chpter 8 APPLICATION OF INTEGRALS One should study Mthemtics ecuse it is only through Mthemtics tht nture cn e conceived in hrmonious form. BIRKHOFF 8. Introduction In geometry,
Prolem 1. f current of 80.0 ma exists in metl wire, how mny electrons flow pst given cross section of the wire in 10.0 min? Sketch the directions of the current nd the electrons motion. Solution: The chrge
Lecture 3 Gussin Probbility Distribution Introduction l Gussin probbility distribution is perhps the most used distribution in ll of science. u lso clled bell shped curve or norml distribution l Unlike
AP STATISTICS SUMMER MATH PACKET This pcket is review of Algebr I, Algebr II, nd bsic probbility/counting. The problems re designed to help you review topics tht re importnt to your success in the clss.
Euler Euler Everywhere Using the Euler-Lgrnge Eqution to Solve Clculus of Vrition Problems Jenine Smllwood Principles of Anlysis Professor Flschk My 12, 1998 1 1. Introduction Clculus of vritions is brnch
Rtio nd Proportion Rtio: The onept of rtio ours frequently nd in wide vriety of wys For exmple: A newspper reports tht the rtio of Repulins to Demorts on ertin Congressionl ommittee is 3 to The student/fulty
MA 5800 Lesson 6 otes Summer 06 Rememer: A logrithm is n eponent! It ehves like n eponent! In the lst lesson, we discussed four properties of logrithms. ) log 0 ) log ) log log 4) This lesson covers more
Cndin Assocition of Physicists 1999 Prize Exm This is three hour exm. Ntionl rnking nd prizes will be bsed on student s performnce on both sections A nd B of the exm. However, performnce on the multiple