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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 257 54 "High School Modul
es > Precalculus by Gregory A. Moore\n" }}{PARA 3 "" 0 "" {TEXT -1 4 "
    " }{TEXT 256 15 "Conic Sections " }{TEXT 277 8 "(Part 1)" }}{PARA 
0 "" 0 "" {TEXT -1 208 "\nAn exploration of three of the conic section
s - the circle, the parabola and the trivial case of intersecting line
s. We will look at the two and three dimensional graphs to see how the
se curves are defined.\n" }}{PARA 0 "" 0 "" {TEXT 258 153 "[Directions
 : Execute the Code Resource section first. Although there will be no \+
output immediately, these definitions are used later in this worksheet
.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 
-1 1 " " }{TEXT 260 7 "0. Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 22 "restart; with(plots): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 239 "sc := 'scaling = constrained':\nt1 := 'thickness = 1':\nt4 := '
thickness = 4':\nl2 := 'linestyle = 2':\nl3 := 'linestyle = 3':\nbl :=
 'color = COLOR(RGB, .4,.5,.7)':\nbl2 := 'color = COLOR(RGB, .2,.25,.3
5)':\nro  := 'color = COLOR(RGB, .9,.6,0)':" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 202 "cone1 := cylinderplot( [r,theta,r],r = 0..5, thet
a=0..2*Pi,\n                   style = patchnogrid):\ncone2 := cylinde
rplot( [r,theta,-r],r = 0..5, theta=0..2*Pi,\n                   style
 = patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 295 "#_____
___ LINES _________________________________________\nline1 :=  spacecu
rve([ t,0,  t], t=-5..5, color = red):\nline2 :=  spacecurve([ t,0, -t
], t=-5..5, color = black):\nplnl :=   implicitplot3d( y = 0 ,x=-5..5,
y=-5..5, z=-5..5,\n                        style=patchnogrid, shading=
ZGRAYSCALE ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 420 "#________ PARABOLA ________
_________________________________\npar :=  spacecurve([ t, 1-(t^2)/4, \+
1+(t^2)/4 ],t=-4..4, \n                      color=blue, thickness=2):
\nplnp :=  plot3d( 2- y, x=-5..5, y= -5..5, \n                      st
yle=patchnogrid, shading=ZGRAYSCALE ):\nxap :=  spacecurve([ t, 1, 1  \+
],t=-5..5, color = COLOR(RGB, .3,.7, .3) ):\nyap :=  spacecurve([ 0, t
, 2-t],t=-3..5, color = COLOR(RGB, .3,.7, .3) ):\n" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 417 "#________ CIRCLES ________________________
_________________\ncir :=  spacecurve([ 3*cos(t), 3*sin(t), 3 ],t=-4..
4, \n                      color=red, thickness=2):\nplnc :=  plot3d( \+
3, x=-5..5, y= -5..5, \n                    style=patchnogrid,  color \+
= COLOR(RGB,.67,.67,.75) ):\nxac :=  spacecurve([ t, 0, 3 ],t=-5..5, c
olor = COLOR(RGB,.2,.2,.6) ):\nyac :=  spacecurve([ 0, t, 3 ],t=-5..5,
 color = COLOR(RGB,.2,.2,.6) ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "pln1;" }}}}{SECT 
0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 263 15 "1. Two Lines   " }
{TEXT 266 38 "  Plane Intersects Cone Through Origin" }{TEXT 267 3 "  \+
 " }}{PARA 0 "" 0 "" {TEXT -1 174 "\nA double cone is a relatively sim
ple three dimensional shape. However, there are several types of curve
s which come about as intersections of the cones and a single plane. \+
\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "display([cone1, cone2]
, orientation = [25,75] );" }}}{PARA 0 "" 0 "" {TEXT -1 159 "\nThis is
 the simplest intersection. If we cut the cones with a single plane wh
ich passes through the axes of symmetry of the cones, we get something
 like this." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "display([cone
1, cone2, line1,line2, plnl], orientation = [25,75] );" }}}{PARA 0 "" 
0 "" {TEXT -1 66 "\nThe intersection of these shapes is a pair of inte
rsecting lines." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "display([
cone1, cone2, line1,line2], orientation = [25,75] );" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 62 "display([cone1, cone2, line1,line2], orie
ntation = [-50,40] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "di
splay([line1,line2, plnl], orientation = [33,71] );" }}}{PARA 0 "" 0 "
" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 
264 12 "2. Parabola " }{TEXT 268 45 "      Plane Intersects Cone Paral
lel to Edge " }{TEXT 269 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 434 "\nNow
 we intersect the plane with the cones in a different way. We are goin
g to tilt the plane so that its \"slope\" is the same as the edge of t
he cone. (For example, we could keep the plane passing through the ori
gin where the cones meet, and tilt it until it becomes tangent to the \+
cones, then raise is straight up or down.) The resulting shape is a pa
rabola. Notices that the plane only intersects one of the two cones in
 this case.\n" }}{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "dis
play([ par, plnp, cone1, cone2,xap,yap ], orientation = [25,75] );" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "display([ par, cone1, cone2
 ], orientation = [25,75] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 55 "display([ par, plnp,xap,yap ], orientation = [35,75] );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "display([ par, plnp,xap,yap \+
], orientation = [ -90,135],axes = none );" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" 
{TEXT -1 1 " " }{TEXT 265 18 "3. Parabola       " }{TEXT 272 27 "Equat
ion & Standard Graph  " }{TEXT 273 19 "                   " }}{PARA 0 
"" 0 "" {TEXT -1 1 "\n" }{TEXT 275 26 "    Plain Vanilla Parabola" }
{TEXT -1 96 "\n\nHere is a very simple parabola and some variations th
at occur when shifting and stretching it." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "plot( x^2, x = -3..3, t4,bl);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 115 "display( plot( (x-3)^2 + 5, x = -3..6,  y = 0
..20, t4,bl),\n         plot( x^2, x = -3..6, y = 0..20, t1,bl2, l2) )
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "display( plot( 3*x^2,
 x = -3..3,  y = 0..20, t4,bl),\n         plot( x^2, x = -3..3, y = 0.
.20, t1,bl2, l2) );" }}}{PARA 0 "" 0 "" {TEXT -1 3 "\n\n\n" }{TEXT 
276 21 "    Formal Definition" }{TEXT -1 152 "\n\nHere is the formal d
efinition of a parabola... as the set of points equidistant between a \+
point, (0,1) in this case, and a line, y = -1 in this case.\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 430 "display( plot( (1/4)*x^2, x
 = -3..3,t4,bl),\n         plot( [[-3,-1],[3,-1]], t4, ro, l3),\n     \+
    pointplot( [0,1], ro, symbol = diamond, symbolsize = 30),\n       \+
  plot( [[-1,-1],[-1,1/4],[0,1]], t1, color = green, l2),\n         pl
ot( [[-2.5,-1],[-2.5,25/16],[0,1]], t1, color = green, l2),\n         \+
plot( [[ 1,-1],[1,1/4],[0,1]], t1, color = green, l2),\n         plot(
 [[ 2.5,-1],[ 2.5,25/16],[0,1]], t1, color = green, l2)  );" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }{TEXT 261 12 "4.
 Circle   " }{TEXT 270 60 "         Plane Intersects Cone Perpindicula
r to Axis of Cone" }{TEXT 271 3 "   " }}{PARA 0 "" 0 "" {TEXT -1 180 "
\nAnother way to intersect the cones is with a plane which is perpindi
cular to the axes of symmetry of the cones - much like chopping a carr
ot with a knife. The result is a circle.\n" }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 60 "display([ cir, plnc, cone1, cone2], orientation = [
33,72] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "display([ cir,
 plnc, cone1, cone2,xac,yac ], orientation = [48,63] );" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "display([ cir, plnc, xac,yac ], ori
entation = [43,62] );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "di
splay([ cir, plnc, xac,yac ], orientation = [90,0] );" }}}{PARA 0 "" 
0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 1 " " }
{TEXT 262 21 "5. Circle            " }{TEXT 274 27 "Equation & Standar
d Graph  " }}{PARA 0 "" 0 "" {TEXT -1 90 "\nCircles are very simple si
nce one looks like another - except for the size and position.\n" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "implicitplot( x^2 + y^2 = 2^
2, x = -3..3, y=-3..3,t4,bl,sc);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 176 "display( implicitplot(  x^2 + y^2 = 2^2,  x = -3..3,
 y=-3..3, t1,bl2, l2),\n         implicitplot( (x-2)^2 + (y-4)^2 = 2^2
, x = -3..6,y=-3..7,\n                       t4,bl,sc));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "display( implicitplot(  x^2 + y^2 \+
= 2^2,  x = -3..3, y=-3..3, t1,bl2, l2),\n         implicitplot( (x+2)
^2 + (y+3)^2 = 3^2, x = -6..6,y=-6..3,\n                       t4,bl,s
c));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "
" {TEXT -1 2 "\n " }}}{PARA 0 "" 0 "" {TEXT 259 35 "\n         \251 20
02 Waterloo Maple Inc" }}}{MARK "0 1" 32 }{VIEWOPTS 1 1 0 1 1 1803 1 
1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }