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{SECT 0 {EXCHG {PARA 208 "" 0 "" {TEXT 200 15 "MAT 156  F'05\n" }{TEXT
 200 7 "Lab 4\n" }{TEXT 200 20 "Circles and Ellipses" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT 206 2 "\n" }{TEXT 206 20 "Draw the graph of   " }
{XPPEDIT 18 0 "x^2+y^2 = 9;" "6#/,&*$)%\"xG\"\"#\"\"\"F)*$)%\"yGF(F)F)
\"\"*" }{TEXT 206 2 "\n" }}{PARA 0 "" 0 "" {TEXT 206 29 "  Method 1 : \+
Solve Explicitly" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "upper:=
 x->sqrt(9-x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&upperGf*6#%\"xG
6\"6$%)operatorG%&arrowGF(-%%sqrtG6#,&\"\"*\"\"\"*$)F'\"\"#F1!\"\"F(F(
F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(upper(x), x= -3.
.3);" }}{PARA 13 "" 1 "" {TEXT 207 0 "" }{TEXT 207 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 206 192 "This does not look like at all like a c
ircle.  The problem is that the scale on the x-axis is different than \+
the one on the y-axis.  To prevent this from happening, we modify the \+
plot command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot(upper
(x), x= -3..3,scaling=constrained);" }}{PARA 13 "" 1 "" {TEXT 207 0 ""
 }{TEXT 207 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 206 105 "This now loo
ks like half a circle.  To get the other half we have to solve explici
tly for the bottom half" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "
lower:= x->-sqrt(9-x^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&lowerGf
*6#%\"xG6\"6$%)operatorG%&arrowGF(,$-%%sqrtG6#,&\"\"*\"\"\"*$)F'\"\"#F
2!\"\"F6F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "plot(lowe
r(x), x= -3..3,scaling=constrained);" }}{PARA 13 "" 1 "" {TEXT 207 0 "
" }{TEXT 207 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 206 21 "Now put them
 together" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "plot(\{upper(x
),lower(x)\}, x= -3..3,scaling=constrained);" }}{PARA 13 "" 1 "" {TEXT
 207 0 "" }{TEXT 207 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 206 27 "Meth
od 2 : Solve Implicitly" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "
with(plots):" }}{PARA 7 "" 1 "" {TEXT 208 51 "Warning, the name change
coords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
63 "implicitplot(x^2+y^2=9, x=-3..3, y=-3..3, scaling=constrained);" }
}{PARA 13 "" 1 "" {TEXT 207 0 "" }{TEXT 207 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT 206 66 "The equation for the circle with center (h,k) and r
adius  R is    " }{XPPEDIT 18 0 "(x-h)^2+(y-k)^2 = R^2;" "6#/,&*$),&%
\"xG\"\"\"%\"hG!\"\"\"\"#F)F)*$),&%\"yGF)%\"kGF+F,F)F)*$)%\"RGF,F)" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT 206 0 "" }}{PARA 0 "" 0 "" {TEXT 203 14 
"Assignment 1: " }{TEXT 206 49 "  Plot the circle with center (-1,2) a
nd radius 4" }}{PARA 0 "" 0 "" {TEXT 206 0 "" }}}{EXCHG {PARA 209 "" 0
 "" {TEXT 200 8 "Ellipses" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 206 0 "" }}
}{EXCHG }{EXCHG }{EXCHG }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "imp
licitplot((x-1)^2/9+(y-3)^2/16=1, x=-2..4, y=-1..7, scaling=constraine
d);" }}{PARA 13 "" 1 "" {TEXT 207 0 "" }{TEXT 207 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 206 113 "This is an ellipse whose center is (1,3
) whose horizontal axis has length 6 and whose vertical axis has lengt
h 8." }}{PARA 0 "" 0 "" {TEXT 206 0 "" }}{PARA 0 "" 0 "" {TEXT 206 
138 "The general formula for an ellipse whose center is (h,k) whose ho
rizontal axis has length 2a and whose vertical axis has length 2b is  \+
   " }{XPPEDIT 18 0 "(x-h)^2/(a^2)+(y-k)^2/(b^2) = 1;" "6#/,&*&),&%\"x
G\"\"\"%\"hG!\"\"\"\"#F)*$)%\"aGF,F)F+F)*&),&%\"yGF)%\"kGF+F,F)*$)%\"b
GF,F)F+F)F)" }}{PARA 0 "" 0 "" {TEXT 206 0 "" }}{PARA 0 "" 0 "" {TEXT 
204 13 "Assignment 2:" }{TEXT 206 97 "   Plot the ellipse whose center
 is (0,-1) whose vertical axis has length 4 and horizontal axis 8" }}
{PARA 0 "" 0 "" {TEXT 206 0 "" }}}{EXCHG {PARA 210 "" 0 "" {TEXT 200 
21 "Completing the Square" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 206 95 "It \+
is useful to convert an equation into the standard form for a circle o
r ellipse if possible." }}{PARA 0 "" 0 "" {TEXT 206 0 "" }}{PARA 0 "" 
0 "" {TEXT 206 37 "For example consider the equation  \n" }{XPPEDIT 18
 0 "9*x^2-36*x+16*y^2-32*y;" "6#,**&\"\"*\"\"\"*$)%\"xG\"\"#F&F&F&*&\"
#OF&F)F&!\"\"*&\"#;F&*$)%\"yGF*F&F&F&*&\"#KF&F2F&F-" }{TEXT 206 18 " =
 92           \n" }{TEXT 206 2 "\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 " with(student):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT
 1 0 29 "left:=9*x^2-36*x+16*y^2-32*y;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "right:= 92;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 40 "newleft:=completesquare(left, \{x, y\});" }}}{EXCHG {PARA 0 "" 0
 "" {TEXT 206 64 "The \{x,y\} tells Maple to complete the square in bo
th variables" }}{PARA 0 "" 0 "" {TEXT 206 0 "" }}{PARA 0 "" 0 "" {TEXT
 206 55 "The next step is to move the -52 to the right hand side" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "right := right +52; finallef
t:= newleft+52;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 206 0 "" }}{PARA 0 ""
 0 "" {TEXT 206 58 "Then divide by 196 to make the right hand side equ
al to 1." }}{PARA 0 "" 0 "" {TEXT 206 0 "" }}{PARA 0 "" 0 "" {TEXT 
206 36 "Finally we give the equation a name." }}}{EXCHG {PARA 0 "> " 0
 "" {MPLTEXT 1 0 26 "eqn:= finalleft/right = 1;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 58 "implicitplot(eqn, x=-6..8,y=-12..16, scaling=c
onstrained);" }}{PARA 13 "" 1 "" {TEXT 207 0 "" }{TEXT 207 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 205 13 "Assignment 3:" }{TEXT 206 26 "   \+
 Identify the equation " }{XPPEDIT 18 0 "9*x^2+4*y^2+36*x = 0;" "6#/,(
*&\"\"*\"\"\"*$)%\"xG\"\"#F'F'F'*&\"\"%F'*$)%\"yGF+F'F'F'*&\"#OF'F*F'F
'\"\"!" }{TEXT 206 56 " as a circle or ellipse in standard form.  Then
 plot it." }}}}
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