7 Parameric equaions This chaer will show ou how o skech curves using heir arameric equaions conver arameric equaions o Caresian equaions find oins of inersecion of curves and lines using arameric equaions differeniae arameric equaions o find equaions of angens and saionar values inegrae arameric equaions o find areas under curves. Before ou sar You should know how o: Subsiue ino formulae. e.g. If a = + and b = -, find when = a - b Subsiue for a and b: = ( + ) - ( - ) = + + 9 - + = + 6 + 8 Check in: If m = ( + ) and n = find when a = m + n b = m + 6n Solve simulaneous equaions. e.g. Solve = + and + 5 = Eliminae : ( + ) + 5 = - - 6 = 0 ( + )( - ) = 0 So, = - or and = - or The soluions are (-, -) and (, ). Differeniae and inegrae funcions. e.g. Find d d and d if = + Eand he brackes: = + + = + + Hence d = ( ) d + ( ) = 5 5 and d = + + + c = + c Solve hese simulaneous equaions. a + =, + = 5 b + =, + = Find d and d when d a = + + b = + + c = + ( )( ) 59
7. Parameric equaions and curve skeching A Caresian equaion has he form = f() e.g. = + + Some relaionshis beween and involve a hird variable. This hird variable is called a arameer. The equaions = f(), = g() are called arameric equaions. is he arameer. You can skech a curve described b arameric equaions b finding oins on he grah for a range of values of. Each oin on he grah has a value of associaed wih i. EXAMPLE Skech he grah of he curve wih he arameric equaions = -, = for - Consruc a able of values: - - - - 0-9 -7-5 - - 5 7 6 9 0 9 6 = 6 = I is useful o label each oin wih is value of. You can hen see how he curve akes shae as varies. = = = = = = = 0 9 7 The curve is a arabola. 60
7 Parameric equaions EXAMPLE Skech he curve given aramericall b he equaions = sin q, = sin q for 0 q Consruc a able of values for 0 q : q 0 0 0.707 0.707 0-0.707 - -0.707 0 0 0-0 0-0 5 7 q is he arameer. q is in radians. i = i = i = 5 i = 0,, 7 i = i = i = As q increases from 0, he curve races he wo loos of a figure-of-eigh. When a curve is eressed using arameric equaions, ou can find he Caresian equaion b eliminaing he arameer (or q). EXAMPLE Find he Caresian equaion of he curves which have hese arameric equaions. a = -, = 8 + b = sin q +, = cos q 5 a Subsiue from = - ino he equaion for : From = -, ( ) = +, so = 8 + = ( + ) - = + + The Caresian equaion is = + + b Find sin q and cos q in erms of and : From = sin q +, sin q = From = cos q - 5, cos q = Subsiue ino sin q + cos q = : + 5 ( ) + ( ) = ( - ) + ( + 5) = + 5 The Caresian equaion is ( - ) + ( + 5) = + + is a quadraic eression, which indicaes ha he curve is a arabola. This equaion reresens a circle, cenre (,-5) and radius =. See C for revision. 6
7 Parameric equaions You can also find arameric equaions of a curve reresened b a Caresian equaion. EXAMPLE Find arameric equaions for he curve wih he Caresian equaion = 6 You need o find a arameer which will simlif Recall ha - sin q = cos q Le = sin q: So = 6sinq sin q = 6sin q cos q = (sin q cos q) = sin q Hence, arameric equaions for he curve are = sin q, = sin q The arameer is q. Leing = cos q will also give = sin q. Tr his ourself. There ma be more han one ossible air of arameric equaions for a given curve. Eercise 7. A curve has he arameric equaions =, = - Co and comlee his able. Hence skech he grah of he curve for - - - - 0 The arameric equaions of a curve are =, = Co and comlee his able. Hence skech he grah of he curve for - - - 0 Consruc our own ables of values o skech he grahs of he curves wih hese arameric equaions for he range of values given in each case. a = -, = for - b = - +, = - for - c =, = for - d = sin q, = cos q for 0 q e = 5cos q, = sin q for 0 q f = sec q, = an q for 0 q 6
7 Parameric equaions Find he Caresian equaion for each of he curves given aramericall b hese equaions. a = + = - b = = c = + = - d = = e = - = + f = = g = = + h = cos q = sin q i = sin q = cos q j = cos q = 5cos q k = sec q = an q l = + 5 Poin P lies on he curve = -, = + If he -coordinae of P is 6, find is -coordinae. = + 6 Poin Q lies on he curve = +, = If he -coordinae of Q is, find is -coordinae. 7 The oin (, k) lies on he curve = - 5, = - Find he ossible values of k. 8 The variable oin P (a, - ) mees he line = 8 a he oin (6, 8). Find he ossible values of a and he Caresian equaion of he curve along which P moves. 9 Find he coordinaes of he oins where hese curves mee he -ais. a = + = - b = + = - c = 5 + = - d = cos q = sin q 0 Find he coordinaes of he oins where hese curves mee he -ais. a = - 5 = - b = - + = + c = - = d = an q = sec q The curve = a -, = a( - ) conains he oin (7, 0). Find he value of a. The oin (0, 0) lies on he curve = a, = a Find he value of a. 6
7 Parameric equaions The curve = asin q, = + acos q inersecs he -ais a he oin (0, ). Find he value of a. Poins A and B lie on he curve = -, = + where = and = resecivel. Find a he disance beween A and B b he gradien of he chord AB c he equaion of he chord AB. 5 Show ha hese wo airs of arameric equaions reresen he same sraigh line. Find he Caresian equaion of he line. a = - b = + = - = + + 6 Find arameric equaions of he curve wih he Caresian equaion = if q is he arameer such ha = cos q 7 Use he ideni + an q = sec q o find arameric equaions for he curve wih he Caresian equaion = + 8 The Caresian equaion of a curve is = Find arameric equaions for his curve if a = sin q b = 9 The equaion of a circle is + - 6 - + = 0 a The equaion is wrien in he form ( - a) + ( - b) = Find he values of a and b. b Hence, find arameric equaions for he circle in erms of he arameer q. Refer o Eamle. 0 A herbola has he equaion 9 - - 8 + 6 - = 0 a The equaion is wrien in he form ( a) ( c ) = b d Find he values of a, b, c and d. b Hence, find arameric equaions for he herbola in erms of q. 6
A curve has arameric equaions = - sin, = - cos, 0 a Find he values of, in erms of, a he wo oins where he curve crosses he -ais. b The curve crosses he -ais a wo oins where = a and = b Show ha one of hese oins has a = 0. Find, b rial-and-imrovemen, he value of b o decimal lace. Find he coordinaes of hese wo oins on he -ais. B subsiuing =, find arameric equaions for he curves wih hese Caresian equaions. a = b = - c - = d - = 7 Parameric equaions A curve has he Caresian equaion + = a Show, b subsiuing =, ha he curve can be reresened b he arameric equaions = =, + + b i Find he oins where = 0 and = ii Invesigae he curve when is close o -. c Hence, skech he curve and find he equaion of an asmoe. INVESTIGATIN a Use a comuer s grahical ackage o check our answers o quesions, and. You can also check answers o oher quesions b drawing aroriae grahs. b Invesigae how changing he values of consans A, B, m and n in hese arameric equaions alers he grahs of he curves. = Asin mq, = Bcos nq for 0 q Your comuer sofware ma need o have q in degrees, ha is, 0 q 60 65
7. Poins of inersecion You can use simulaneous equaions o find he oins of inersecion when a curve is eressed in arameric equaions. EXAMPLE Find he oins of inersecion of he curve wih arameric equaions = -, = and he sraigh line = - Solve he equaions = -, = simulaneousl. = - Subsiue = -, = ino = - o eliminae and : = ( - ) - - 6 + 5 = 0 ( - )( - 5) = 0 so = or 5 When =, = () - = and = = When = 5, = (5) - = 9 and = 5 = 5 There are hree unknowns,, and. The oins of inersecion are (, ) and (9, 5). EXAMPLE Find he oins A, B and C where he curve given aramericall b = -, = - inersecs he wo coordinae aes. Hence, find he area of riangle ABC. When = 0, - = 0 giving = When =, = - = -, giving he oin of inersecion A(-, 0). When = 0, - = 0 giving = ± When =, = - = and, when = -, = - - = -, giving he oins of inersecion B(0, ) and C(0, -). The area of riangle ABC is BC A = = 6 square unis The curve mees he -ais when = 0 The curve mees he -ais when = 0 B A C 66 Eercise 7. Find he oins of inersecion of he arabola = = and he sraigh lines a + = b + = 5
Find he oins of inersecion of each curve and he given line. a = -, = + = + b =, = + = + c = -, = + + - - = 0 Find he oins of inersecion of he curve wih arameric equaions =, = and he circle + - 6 - = 0 Find he oins of inersecion of he arabola + = 9 and he curve = ( - ), = 5 Find he oins where hese curves cross he coordinae aes. 7 Parameric equaions a = - = - b = - = - 9 c = + = - d = - = - e = 6 The variable oin P(, ) moves along a locus. Find he oins where he locus crosses he sraigh line = - = + f = - = - sin 7 The oin P(, ) moves as varies. Q is he midoin of P where is he origin. Wrie down he coordinaes of Q. Find he Caresian equaion of he locus of Q. 8 The oin P(, 6) lies on a curve. The foo of he erendicular from P o he -ais is Q. The midoin of PQ is M. Find a he coordinaes of Q and M in erms of b he Caresian equaion of he locus of M as P moves. 9 Find he oins of inersecion of he curve = - 5, = + and he line + + = 0 0 The curve = +, = - k inersecs he - and -aes a oins P and Q resecivel. Find he value of k (k ¹ ) such ha P = Q where is he origin. INVESTIGATIN Use a comuer s grahical sofware o draw grahs using heir arameric equaions. Check our answers o he roblems in his eercise where ou have found oins of inersecion. 67
7. Differeniaion You can differeniae arameric equaions o obain d d. If = f() and = g() he chain rule gives d d d d = or d d d d = d d d d nce ou know d, ou can use i o find equaions of angens d and normals o a curve and o find saionar values. EXAMPLE A curve has arameric equaions = + +, = - Find a he equaion of he angen a he oin where = b he naure of an saionar values and he oins a which he occur. a Differeniae wr : Differeniae wr : So d = + d d = d d d d = d = d + d Subsiue = : = = + When = = + + = 6 and = - = So, he angen asses hrough he oin (6, ) wih a gradien of 7. The equaion of he angen is = 6 7 Rearrange: 7 = - 7 is he gradien of he 7 angen when = = m See C for revision. d b For saionar values, = = d 0 + So he onl saionar value occurs when = 0 a he oin where = 0 + 0 + = and = 0 - = - - 0 7 d d 5 0 5 The numeraor = 0 when = 0 Invesigae he gradien on eiher side of he oin (, -): Choose values of eiher side of = 0 and make sure ha he -values are eiher side of = 68 There is a minimum value a he oin (, -). 6
7 Parameric equaions EXAMPLE A curve is defined aramericall b = -, = - - A normal is drawn o he curve a he oin A where = Find anoher oin B a which his normal inersecs he curve. Le = : = - = and = - - = So, he normal is drawn a he oin A (, ). Find d and subsiue = : d d d d = = d d d = + When =, he gradien of he angen a he oin A(, ) is + () = The gradien of he normal a he same oin is -. So, he equaion of he normal a he oin A(, ) is = + = 8 To find he inersecions of he normal and curve, subsiue he arameric equaions ino + = 8: ( - - ) + ( - ) = 8 + - 7 = 0 ( - )( + 7) = 0 = or - 7 = gives he iniial oin A on he normal, If he gradiens of angen and normal are m and m, hen m = - m You know ha he curve and normal inersec when = Hence a B, = - 7 Subsiue = 7 ino he arameric equaions: = + 7 = 5 = + 7 9 = 8 9 9 The required oin of inersecion is 5 8 (, 9). A B 6 69
7 Parameric equaions Eercise 7. Find he gradien of each curve wih hese arameric equaions a he oin wih he given value of (or q). a = + = when = b = - = + when = c = - = ( + ) when = d = sin q = 5cos q when q = e = + = + when = - f = sin q = qcos q when q = 0 Find he equaion of he angen and he normal o he curves wih hese equaions a he oin where (or q) has he given value. a = = when = b = + = - when = c = cos q = cos q when q = d = + = + when = Find he saionar oins on hese curves. a = = - b = = + c = q - cos q = sin q for 0 < q < d = sin q + = cos q + 5 for 0 q The curve =, = has a normal a oin P(8, 8). Find he equaion of he normal. Also find he oin where he normal mees he curve a second ime. 5 Find he equaion of he normal o he curve = 6, he oin where = = 6 a Also find he oin where he normal inersecs he curve again. 70
7 Parameric equaions 6 The angen a oin P(, ) o he curve he curve a oin Q. =, = inersecs Find he equaion of he angen a P and he coordinaes of Q. 7 The oin P lies on he curve = 5cos q, = sin q A and B are he oins (-, 0) and (, 0) resecivel. a Find he disances AP and BP in erms of q. b Show ha he sum of he disances AP and BP is consan for all oins P. ( ) 8 The line from he variable oin P, inersecs he line = a he oin Q. = o he origin Q P Find a he gradien and he equaion of he line P b he coordinaes of Q in erms of c he Caresian equaion of he locus of he midoin of PQ. INVESTIGATIN 9 Invesigae how o draw angens and normals o curves using a comuer s grahical ackage. Hence, check some of our answers o he roblems in his eercise. 7
7. Inegraion You can modif he eression curve using he chain rule. a b f() d for he area under a For arameric equaions = f(), = g(), he area under he curve beween he oins where = and = is given b a b d d d The limis of his inegral are = and =, as he indeenden variable is now and no. EXAMPLE This skech shows he curve wih arameric equaion = +, = + Find he shaded area beween he curve and he -ais from = o = 0 = = B A 0 Le = : =, = ± so = or -. So, oin A is (, ). Le = 0: = 9, = ± so = or -. So, oin B is ( 0, ). You wan he area under he curve from A(, ) where = ( ) where = o B 0, You are calculaing he area above he -ais onl. 7 Inegrae: Required area = = 0 d = ( ) + () d = ( + ) d = + ( ) = 9+ d d d where = + and d d = = square unis Noice he change in he limis as he indeenden variable changes from o.
7 Parameric equaions EXAMPLE This diagram shows he curve wih arameric equaions = +, = - Find he values of a he oins A(, 0) and B(5, 0). Find he area of he region enclosed b he loo of he curve. = B A = ± = 0 5 = Find a A and B: A oins A and B, = 0 - = 0 ( - )( + ) = 0 = 0 or ± Le = 0: = 0 + = and = 0-0 = 0 So, = 0 a he oin A(, 0). Find and when = 0 Le = : = + = 5 and = - = 0 Le = -: = (-) + = 5 and = (-) - (-) = 0 So, = ± a he oin B(5, 0). Find and when = + and = -. You wan he area under he curve from A(, 0) where = 0 o B(5, 0) where = ± When =, =, = - = -, giving he oin (, -) below he -ais. So, inegraing from = 0 o = will give he area below he -ais. Similarl, = - gives he oin (, ) and inegraing from = 0 o = - gives he area above he -ais. area of loo = area enclosed b he curve above he -ais = 5 d = 0 d d d where = - and d = d The curve is smmerical abou he -ais. = = 0 0 ( - )()d ( - )d = 5 5 0 ( ) = + 0 5 = 7 square unis 5 You could find he answer direcl b inegraing he whole wa from = - o = using - 8 d Tr his ourself. 7
7 Parameric equaions Eercise 7. Find, in each case, he area beween he -ais and he curve beween he wo oins P and Q defined b he given values of or. a = + = - for = o = b = = + for = 0 o = Q c = = + - for = 0 o = d = - = - for = o = 5 P e = = + for = o = f = = for = o = A and B are he oins on he curve =, = where = 0 and = Find he shaded area on he diagram. Also find he area of he region labelled R. A R = 0 B = a The curve = -, = ( - ) cus he coordinae aes a he oins A, B, C and D. Find he osiions of hese oins and heir associaed -values. b Calculae he area of he region R in he firs quadran enclosed b he curve and he wo coordinae aes. C B R A c Calculae he oal shaded area. D This diagram shows he curve =, = ( - ) Find he values of a he wo oins where he curve cus he -ais. Hence, find he area enclosed b he loo. 7
5 a This diagram shows he curve = +, = Find he coordinaes of he oins A, B and C on he curve where =, and resecivel. 7 Parameric equaions C b Calculae he shaded area of he diagram bounded b he curve and he line AC. B A 6 The curve = - +, = + is shown on his diagram. Find he oins a which he curve cus he -ais. Also find he area bounded b he curve and he line = 5 7 a A curve is eressed aramericall b = +, = Anoher curve has he arameric equaions = s, = s Find an oins of inersecion. b Find he area enclosed b he wo curves and he line = 5 8 a The sraigh line = c - ouches he curve =, = a oin P. Find he ossible values of c and he coordinaes of P. b Find he area enclosed b he curve =, = line = - and he INVESTIGATIN 9 Show ha, when finding he area under a curve, ou ge he same answer wheher ou use he curve s Caresian equaion or is arameric equaions. Consider some of he curves in quesions and of his eercise. 75
Review 7 Find he Caresian equaion for each of he curves given aramericall b hese equaions. a = -, = + b =, = + c = cos q, = sin q d = cos q, = cos q Use he ideni sin q + cos q º o find arameric equaions for he curves wih he Caresian equaion a = b = 5 The oin (5, a) lies on he curve = +, = ( ) Find he ossible values of a. Find he oins where he curve given b hese arameric equaions a = +, = - inersecs he sraigh line + = 6 b = -, = - inersecs he -ais c = cos, = 5sin inersecs he circle + = 5 Find he equaion of he angen and he normal o he curves wih hese arameric equaions a he oin where has he given value. a = -, = + when = b = -, = + + when = c = +, = when = d = sin, = sin when = 6 6 Find, in each case, he area beween he -ais and he curve beween he wo oins defined b he given values of or. a = + from = o = b = + from = o = 9 = - = c = ln from = o = d = e - from = o = = sin = e + 76 7 The curve C is defined aramericall, for 0 q, b he equaions = cos q, = cos q + 6 a Find d in erms of q. d Elain wh he gradien a an oin on he curve C is never greaer han. b Find he Caresian equaion of C and skech he grah of C.
7 Parameric equaions 8 A curve has arameric equaions =, = R is he oin on he curve where = r a Show ha he normal o he curve a oin R has a gradien of -r. b If S is he oin (s, s), find he gradien of he chord RS in erms of r and s. c If he chord RS is normal o he curve a R, show ha r + rs + = 0 d A wha oin does he normal a he oin (9, 6) mee he curve again? 9 The rajecor of a cricke ball is given aramericall b he equaions = 0, = + 0-5 w h ere and are he horizonal and verical disances ravelled (in meres) afer a ime of seconds from he ball being sruck. a b c Find he Caresian equaion of he rajecor. Find he ime aken before he ball his he ground. Wha is he horizonal disance ravelled b he ball before i his he ground? 0 This diagram shows a skech of ar of he curve C wih arameric equaions = +, = sin, a The oin P a, lies on C. Find he value of a. b Region R is enclosed b C, he -ais and he line = a as shown in he diagram. C R a P Show ha he area of region R is given b c Find he eac value of he area of R. sin d A curve has arameric equaions a b c = co q, = sin q, 0 < q Find an eression for d in erms of he arameer q. d Find an equaion of he angen o he curve a he oin where q = Find a Caresian equaion of he curve in he form = f() Sae he domain on which he curve is defined. [(c) Edecel Limied 005] 77
7 Ei Summar Refer o You can skech he grah of a curve given b he arameric equaions = f(), = g() b using a able of values showing he values of and as varies. 7. To conver he arameric equaions = f(), = g() of a curve ino a Caresian equaion, eliminae he arameer from he wo equaions. 7. You can find he oins of inersecion of wo curves b solving he equaions of he curves simulaneousl. For eamle, ou can subsiue he arameric equaions of one curve ino he Caresian equaion of he oher curve. 7. To differeniae = f(), = g() o find d d, he chain rule gives: d d d g d = d = () f 7. () = b d = d The area under a curve is given b d = d d = a = where he arameer has he value a he oin where = a and he value a he oin where = b 7. Links The ah of an rojecile is he resul of wo indeenden moions, horizonal and verical, which can be eressed in erms of a arameer ime. You can model he ah of a rojecile a an ime b he arameric equaions () = (n 0 cos q), () = (n 0 sin q) g where q is he angle a which he rojecile is launched a ime = 0, n 0 is he iniial veloci of he rojecile, and g is he acceleraion due o gravi. This model can be used o analse he moion of a secific rojecile, which is useful in areas such as sors science o sud, for eamle, he fligh of a golf ball. 78