{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "" -1 256 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 257 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 263 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 128 0 1 0 2 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 269 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 128 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 128 0 128 1 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 275 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 128 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE " Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 256 54 "High School Modul es > Precalculus by Gregory A. Moore\n" }}{PARA 3 "" 0 "" {TEXT -1 4 " " }{TEXT 266 15 "Conic Sections " }{TEXT 267 8 "(Part 2)" }}{PARA 0 "" 0 "" {TEXT -1 169 "\nAn exploration of two more of the conic sect ions - the ellipse and hyperbola. We will look at the two and three di mensional graphs to see how these curves are defined.\n" }}{PARA 0 "" 0 "" {TEXT 257 153 "[Directions : Execute the Code Resource section fi rst. Although there will be no output immediately, these definitions a re used later in this worksheet.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 259 7 "0. Code" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart; with(plots): " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 239 "sc := 'scaling = constraine d':\nt1 := 'thickness = 1':\nt4 := 'thickness = 4':\nl2 := 'linestyle \+ = 2':\nl3 := 'linestyle = 3':\nbl := 'color = COLOR(RGB, .4,.5,.7)':\n bl2 := 'color = COLOR(RGB, .2,.25,.35)':\nro := 'color = COLOR(RGB, . 9,.6,0)':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 202 "cone1 := cyli nderplot( [r,theta,r],r = 0..5, theta=0..2*Pi,\n sty le = patchnogrid):\ncone2 := cylinderplot( [r,theta,-r],r = 0..5, thet a=0..2*Pi,\n style = patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 574 "#________ ELLIPSES _________________________________________\np lne := implicitplot3d( z = 2 - (1/2)*y, x=-5..5, y= -5..3,z=-5..5, \n style=patchnogrid, shading=ZGRAYSCALE ):\n \+ \nell := spacecurve( [ (4/sqrt(3))*cos(t), \n \+ (8/3)*sin(t)-4/3, \n 2-(1/2 )*((8/3)*sin(t)-4/3), t = 0..2*Pi], \n color = COLOR(RGB, . 9,.8,.2),thickness = 3 ):\nxae := spacecurve([ t, 0, 2 ],t=-5..5, colo r = COLOR(RGB,.2,.2,.6) ):\nyae := spacecurve([ 0, t, 2-(1/2)*t],t=-5. .3,color=COLOR(RGB,.2,.2,.6)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "#________ HYPERBOLAS _________________________________________ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 488 "plnh := implicitplot 3d( y = 2 ,x=-5..5,y=-5..5, z=-5..5,\n style=pa tchnogrid, shading=ZGRAYSCALE ):\nhyp := spacecurve( [ t, 2, sqrt(4+t^ 2), t = -5..5], \n color = COLOR(RGB, .9,.8,.2),thickness = 3 ):\nhyp2 := spacecurve( [ t, 2, -sqrt(4+t^2), t = -5..5], \n \+ color = COLOR(RGB, .9,.8,.2),thickness = 3 ):\nxah := spacecurve ([ t, 2, 0 ],t=-5..5, color = COLOR(RGB,.2,.2,.6) ):\nyah := spacecur ve([ 0, 2, t ],t=-5..5, color = COLOR(RGB,.2,.2,.6)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 260 11 "1. Ellipse " }{TEXT 268 50 " Plane In tersects Cone @ At Shallow Angle" }{TEXT 269 3 " " }{TEXT 264 22 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 291 "\nIf we cut a co ne with a plane which is perpindicular to its axis of symmetry, we get a circle. However, if our cut is not quite perpindicular ... in fact, the \"slope\" of the plane is anything bigger than 0, and less than t he slope of the cone's edge... we can not a circle, but an ellipse.\n " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "display([ ell, plne, con e1, cone2 ], orientation = [6, 90] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "display([ ell, plne, cone1, cone2 ], orientation = [4 6,72] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "display([ ell, \+ plne, cone1, cone2,xae,yae ], orientation = [42,55] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "display([ ell, plne, xae,yae ], ori entation = [57,59] );" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 262 21 "2. Ellipse " } {TEXT 272 25 "Equation & Standard Graph" }{TEXT 273 5 " " }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 52 "Here is a ver y simple ellipse and shifted variant. \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "implicitplot( (x/5)^2 + (y/2)^2 = 1, x = -5..5, y = - 4..4, t4,bl,sc);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 146 "displa y(implicitplot( ((x-3)/5)^2+((y-3)/2)^2=1,x =-3..8,y=0..8,t4,bl),\n \+ implicitplot( (x/5)^2+(y/2)^2=1,x =-5..5,y=-4..4,t1,bl2,l2,sc) \n );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "implicitplot( (x/4)^2 + (y/6)^2 = 1, x = -5..5, y = -6..6, t4,bl,sc);" }}}{PARA 0 "" 0 "" {TEXT -1 3 "\n\n\n" }{TEXT 274 21 " Formal Definition" }{TEXT -1 198 "\n\nAn ellipse is defined in this way. We start with any two poin ts, which are called focus points. The sum of the distances from a poi nt to each of the two fixed focus points (or foci), is constant.\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 399 "f := sqrt(5^2 - 2^2):\ndisp lay( implicitplot( (x/5)^2+(y/2)^2=1,x=-5..5,y=-4..4,t4,bl),\n \+ plot( [[-f,0],[f,0]], t4, ro, l3),\n pointplot( \{[f,0],[-f,0 ]\}, ro,symbol=diamond,symbolsize=30),\n plot( [[-f,0],[3,8/5] ,[f,0]], t1, color = green, l2),\n plot( [[-f,0],[5/2,-sqrt(3) ],[f,0]], t1, color = gray, l2),\n plot( [[-f,0],[-4,6/5],[f,0 ]], t1, color = red, l2), sc);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 497 "f := sqrt(6^2 - 4^2):\ndisplay( implicitplot( (x/4)^2+(y/6)^2 =1,x=-4..4,y=-6..6,t4,bl),\n plot( [[0,-f],[0,f]], t4, ro, l3) ,\n pointplot( \{[0,-f],[0,f]\}, ro,symbol=diamond,symbolsize= 30),\n plot( [[0,-f],[2, sqrt(27)],[0,f]], t1, color = green, \+ l2),\n plot( [[0,-f],[-3/2,sqrt(55)*3/4],[0,f]], t1, color = g ray, l2),\n plot( [[0,-f],[1/2,-sqrt(7)*9/4],[0,f]], t1, color = yellow, l2),\n plot( [[0,-f],[-7/2,-sqrt(15)*3/4],[0,f]], t 1, color = red, l2), sc);" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 261 13 "3. Hyperbola " } {TEXT 271 42 " Plane Intersects Cone @ At Steep Angle" }{TEXT 270 5 " " }{TEXT 265 15 " " }}{PARA 0 "" 0 "" {TEXT -1 401 "\nA hyperbola is created by having a plane intersect a cone where the plane is parallel to the axis of symmetry of the cone. We saw one example of this kind of thing already - when the plane passes through the axis of symmetry - we get two intersecting lines. If the plane is parallel to any plane passing through the axis of symmetry, we get a \+ hyperbola, which is a pair of opposite but equal curves.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "display([ plnh, cone1, cone2 ], ori entation = [156,77] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "d isplay([ hyp, hyp2, plnh, cone1, cone2 ], orientation = [142,75] );" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "display([ hyp, hyp2, plnh, cone1, cone2,xah,yah ], orientation = [142,75] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "display([ hyp, hyp2, plnh, cone1, cone2,xah ,yah],orientation = [42,55]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "display([ hyp,hyp2, plnh, xah,yah ], orientation = [67,85] );" } }}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 275 17 "4. Hyperbola " }{TEXT 276 25 "Equation & S tandard Graph" }{TEXT 277 5 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "\nH ere are some basic hyperbolas.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "implicitplot( (x)^2 - (y)^2 = 1, x = -4..4, y = -4..4, t4,bl,s c);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "implicitplot( (x/5)^ 2 - (y/3)^2 = 1, x = -10..10, y = -7..7, t4,bl,sc);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "implicitplot( -(x/5)^2 + (y/3)^2 = 1, x = -10..10, y = -7..7, t4,bl,sc);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }{TEXT 278 21 " Formal Definition" }{TEXT -1 157 "\n\nA hyperbola is defined in similarly to how an ellipse is d efined - except it's the difference of the two distances rather than t he sum which is constant. \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 277 "f := sqrt(3^2 + 2^2):\ndisplay( implicitplot( (x/3)^2-(y/2)^2=1,x =-6..6,y=-4..4,t4,bl),\n plot( [[-f,0],[f,0]], t4, ro, l3),\n \+ pointplot( \{[f,0],[-f,0]\}, ro,symbol=diamond,symbolsize=30), \n plot( [[-f,0],[5,8/3],[f,0]], t1, color = green, l2),\n \+ sc);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 260 "display( impl icitplot( (x/3)^2-(y/2)^2=1,x=-6..6,y=-4..4,t4,bl),\n plot( [[ -f,0],[f,0]], t4, ro, l3),\n pointplot( \{[f,0],[-f,0]\}, ro,s ymbol=diamond,symbolsize=30),\n plot( [[-f,0],[-6,2*sqrt(3)],[ f,0]], t1, color = red, l2),\n sc);" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 263 40 "5. Ellipses & Hyperbolas on Same Graph " }}{PARA 0 "" 0 "" {TEXT -1 145 "\nIf we draw the two hyperbolas on the same graph, which differ o nly in their signs along with their asymptotes, we get this interestin g diagram.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 331 "display(imp licitplot( (x/5)^2 - (y/3)^2 = 1, \n x = -10..10 , y = -7..7, t4,bl,sc),\n implicitplot( -(x/5)^2 + (y/3)^2 = 1 , \n x = -10..10, y = -7..7, t4,bl,sc),\n \+ plot( (3/5)*x, x = -10..10, l3, color = green), \n plot( -( 3/5)*x, x = -10..10, l3, color = green), sc);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 524 "display(implicitplot( (x/5)^2 - (y/3)^2 = 1, \n x = -10..10, y = -7..7, t4,bl,sc),\n im plicitplot( -(x/5)^2 + (y/3)^2 = 1, \n x = -10..1 0, y = -7..7, t4,bl,sc),\n implicitplot( (x/5)^2 + (y/3)^2 = 1, \n x = -10..10, y = -7..7, t4,ro,sc),\n \+ plot( [[-5,-3],[5,-3],[5,3],[-5,3],[-5,-3]], t1, \n colo r=black, l3),\n plot( (3/5)*x, x = -10..10, l3, color = green) , \n plot( -(3/5)*x, x = -10..10, l3, color = green), sc);" }} }{EXCHG }}{PARA 0 "" 0 "" {TEXT 258 35 "\n \251 2002 Waterloo \+ Maple Inc" }}}{MARK "0 1" 46 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }