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Coord

Core4

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ซานต้า สมศักดิ์
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61 views13 pages

Coord

Core4

Uploaded by

ซานต้า สมศักดิ์
Copyright
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After completing this chapter you should be able to: * sketch the graph of a curve given its parametric equation * use the parametric equation of a curve to solve various problems including the intersection of a line with the curve * convert the parametric equation to a Cartesian form MA © find the area under a curve whose equation is expressed in parametric form. Some very simple-looking curves are very difficult to describe by a Cartesian equation, that is, an equation linking x andy. Parametric equations can be used to produce some very beautiful and Many ancient bridges were reputedly designed on intricate curves. This chapter will the cyclold, which can be easly represented by the introduce you to the topic which parametric equation sit a5) will be studied further in Chapter 4. Coordinate geometry in the (x, 9) plane Gee Draw the curve given by the parametric equations x ~ 2t, y = &, for -3 Acircle has parametric equations x = 4 sin t+ 3, y = 4cost— 1. Find the radius and the coordinates of the centre of the circle. ee 2.4 You need to be able to find the area under a curve given by parametric equations. Ml The area under a graph is given by {y dx. By the chain rule [y dx = [y &= od Ea! a A curve has parametric equations x = 5¢, y = ®. Work out [’ at. at yap __} indy & substitute y =e. Ya as ae ax_d a a _ 4 oeea a 5 nae) me out S Here = 50. Sone St) = 5 x 2 dk =10t So y= Px 10 =10r* ——_____|__ Simplify the expression so that on Px Tot = 100"? 2 = 100 tort dt ; " Integrate, so that x “4 Ph froreae = 29 1b} = 2(2) — 21) os = 64-2 = 285 =62 Work out ee Substitute t= 2 and into 2t° and subtract. ‘The diagram shows a sketch of the curve with parametric equations x =, y = 23-0), t= 0. The curve meets the x-axis at x = 0 and x= 9. The shaded region R is bounded by the curve and the x-axis. a Find the value of ¢ when ix-0 iix-9 b Find the area of R. S0 ii 2 b Area of R =| yax lo ————— =f yXat IV ae 3 =| 28-9 x 2ede-— o = [e- 20") x 2tdt Lh 2 =| 12-4 ae o Is = [ee - | Le = [4(3) — (3)"] — [40° — (0)4] = (108 — 81) - (0-0) ag The area of R= 27. Coordinate geometry in the (x, 9) plane Substitute x = 0 into x = Take the square root of each side. Substitute x = 9 into x= 2. - Take the square root of each side. v9 = +3. Bas Ignore t= Integrate parametrically. Change the limits of the integral t=O when x=0 t= 3.when x=9 Find fy Baw Substitute y = 2t(3 — 0). dx dp a ae) =2 Expand the brackets, so that @ 26-9 =2tx3-2txt =6t-2° @ (t-2e%)x2t =6tx 2t- 2x 2e =122-48 Integrate each term, so that ® fraede=Be4 oe ® fatar -s00 =t Work out [ee = et Substitute t= 3 and t= 0 into 40 — ft and subtract. ‘CHAPTER 2 Exercise 1) The following curves are given parametrically. In each case, find an expression for y & in terms of ¢. a x=tt3,y=4t-3 bx=f+3,y=e © x= (2t-3),y ax-6-hy~ 48, 1>0 ex=Viy=6t,t=0 fx-Zy-set<0 gx=St,y=4t7,t>0 hexe=f'-1,y=\ite0 i x=16-ty=3 2 t<0 jx=6t,y=P 4 2) A curve has parametric equations x = 2t~ 5, y = 3t +8. Work out [ yan lo s 3) A curve has parametric equations x =~ 3t+ 1, y= 4f2, Work out [ y s at. 1 1 3 de 4) A curve has parametric equations x = 31%, y= + #, t>0. Work out | y-T-dt. fos 2 5) A curve has parametric equations x= ~ 4t,y = & — 1, Work out [ y at 6 Acurve has parametric equations x = 913, y= 7, t>0. a Show that y= =a, where a is a constant to be found. Hae 5 ; b Work out | a I 7) Acurve has parametric equations x = Vi, y = 4V#, t> 0. a Show that ye = pt, where p is a constant to be found. Har 6 b Work out I xe 8 The diagram shows a sketch of the curve with parametric equations x = f? - 3, y = 3f, t> 0. The shaded region R is bounded by the curve, the x-axis and the lines x= 1 and x= 6. a Find the value of fwhen ix=1 ii x-6 b Find the area of R. Coordinate geometry inthe (x, 9) plane 9) The diagram shows a sketch of the curve with x parametric equations x = 4, y= 1(S 2), = 0, The shaded region R is bounded by the curve and the x-axis. Find the area of R. 0 25 1 10 The region R is bounded by the curve with parametric equations x = ®, y = 35, the x-axis and the lines x = ~1 and x = -8. a Find the value of tf when ix--1 iix--8 b Find the area of R. 1. The diagram shows a sketch of the curve with parametric equations x = 4 cost, y=3sint, O0 idk <0 © Calculate the area of the finite region enclosed by the curve and the y-axis. Find the area of the finite region bounded by the curve with parametric equations x=B,y £ t #0, the x-axis and the lines x = 1 and x= 8. ‘The diagram shows a sketch of the curve with y parametric equations x = 3Vt, y = t(4 — f), where 0

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