{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 "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Tim es" 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 "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE "Heading 2" -1 256 1 {CSTYLE "" -1 -1 "Ti mes" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 11 "Calculus II" }}{PARA 256 "" 0 "" {TEXT -1 26 "Lesson 24: Conic Sections" }}}{SECT 0 {PARA 3 " " 0 "" {TEXT 260 38 "Geometric Definition of Conic Sections" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 157 "We start with the geome tric definition of conic sections. Later in this lesson, we'll plot c onic sections in the plane using their analytic representations." }} {PARA 0 "" 0 "" {TEXT -1 188 "For each picture below, the conic sectio n is the intersection of the cone with the plane. Grab the picture wi th the mouse and rotate it to see the intersection from many points of view. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "restart: with(s tudent):with(plottools): with(plots):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 6 "Circle" }{TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 248 "up := cone([.3,0,-1],1,2,c olor=blue):\ndown := cone([.3,0,-1],1,-2,color=blue):\ncirclePlane := \+ plot3d(0, x=-3..1.5, y=-1..1, color=green, style=patchnogrid):\ndispla y([circlePlane,up,down], scaling=constrained, orientation=[140,70], st yle=wireframe);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 256 7 "Ellipse" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 176 "ellipsePlane := plot3d(.75*x, x=-3 ..1.5, y=-1..1, color=green, style=patchnogrid):\ndisplay([ellipsePlan e, up,down], scaling=constrained, orientation=[140,70], style=wirefram e);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 8 "Parabola" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "parabolaPlane := plot3d(4*x, x=-.75..(.25), y=- 1..1, color=green, style=patchnogrid):\ndisplay([parabolaPlane, up,dow n], scaling=constrained, orientation=[140,70], style=wireframe);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 258 9 "Hyperbola" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "HyperbolaPlane := implicitplot3d(x=0, x=-.6..(.2), y =-1..1, z=-3..1,color=green, style=patchnogrid):\ndisplay([HyperbolaPl ane, up,down], scaling=constrained, orientation=[140,70], style=wirefr ame);" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 261 41 "Analytic Representati on of Conic Sections" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 262 9 "Example 1" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 97 "Consider the \+ following conics. First try to identify them. Then confirm your gues s by plotting." }}{PARA 0 "" 0 "" {TEXT -1 4 "a) " }{XPPEDIT 18 0 "x^ 2 + y^2 = 9" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(\"\"*" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 4 "b) " }{XPPEDIT 18 0 "x*y = -4" "6#/*& %\"xG\"\"\"%\"yGF&,$\"\"%!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "c) " }{XPPEDIT 18 0 "4*x^2 + y^2 = 16 " "6#/,&*&\"\"%\"\" \"*$%\"xG\"\"#F'F'*$%\"yGF*F'\"#;" }}{PARA 0 "" 0 "" {TEXT -1 4 "d) \+ " }{XPPEDIT 18 0 "x^2 - 4y^2 = 36" "6#/,&*$%\"xG\"\"#\"\"\"*&\"\"%F(*$ %\"yGF'F(!\"\"\"#O" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "e) \+ " }{XPPEDIT 18 0 "y^2 = 4*x " "6#/*$%\"yG\"\"#*&\"\"%\"\"\"%\"xGF)" }} {PARA 0 "" 0 "" {TEXT -1 4 "f) " }{XPPEDIT 18 0 "x^2 + 3*y^2 - 1 = 0 " "6#/,(*$%\"xG\"\"#\"\"\"*&\"\"$F(*$%\"yGF'F(F(F(!\"\"\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "g) " }{XPPEDIT 18 0 "x^2 + 3*x *y + y^2 = 16 " "6#/,(*$%\"xG\"\"#\"\"\"*(\"\"$F(F&F(%\"yGF(F(*$F+F'F( \"#;" }}{PARA 0 "" 0 "" {TEXT -1 4 "h) " }{XPPEDIT 18 0 "x^2 + 4*y = \+ 4 " "6#/,&*$%\"xG\"\"#\"\"\"*&\"\"%F(%\"yGF(F(F*" }}{PARA 0 "" 0 "" {TEXT -1 63 "In each case, state the conic obtained along with 'specif ics'. " }}{PARA 0 "" 0 "" {TEXT -1 36 "Circle specifics = center and r adius" }}{PARA 0 "" 0 "" {TEXT -1 43 "Ellsipse specifics = center, foc i, vertices" }}{PARA 0 "" 0 "" {TEXT -1 45 "Parabola specifics = verte x, directirx, focus" }}{PARA 0 "" 0 "" {TEXT -1 56 "Hyperbola specific s = center, vertices, foci, asymptotes" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "a) Graph is a circle. " }} {PARA 0 "" 0 "" {TEXT -1 20 "Center is at (0,0). " }}{PARA 0 "" 0 "" {TEXT -1 12 "Radius = 3. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "implicitplot( x^2 + y^2 = 9, x = -10..10, y = -10..10, scaling=con strained, grid=[40,40]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "b) Graph is a rotated hyperbola. " }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 63 "implicitplot(x*y = -4, x = -10..10, y = -10..1 0, grid=[40,40]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "c) Graph is an ellsipse. " }}{PARA 0 "" 0 "" {TEXT -1 13 "Center (0,0) " }}{PARA 0 "" 0 "" {TEXT -1 14 "Foci: (0, +- " }{XPPEDIT 18 0 "sqrt(12)" "6#-%%sqrtG6#\"#7" }{TEXT -1 1 ")" }}{PARA 0 "" 0 "" {TEXT -1 30 "major axis vertices: (0,+- 4) " }}{PARA 0 "" 0 "" {TEXT -1 31 "minor axis vertices: (+-2, 0)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "implicitplot(4* x^2 + y^2 = 16, x = -10..10, y = -10. .10, grid=[50,50]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "d) Graph is a hyperbola. " }}{PARA 0 "" 0 "" {TEXT -1 11 "Cent er(0,0)" }}{PARA 0 "" 0 "" {TEXT -1 19 "Vertices: (+- 6, 0)" }}{PARA 0 "" 0 "" {TEXT -1 10 "Foci( +- " }{XPPEDIT 18 0 "3*sqrt(5)" "6#*&\" \"$\"\"\"-%%sqrtG6#\"\"&F%" }{TEXT -1 4 ", 0)" }}{PARA 0 "" 0 "" {TEXT -1 20 "asymptotes: y = +- " }{XPPEDIT 18 0 "(1/2) *x" "6#*(\"\" \"F$\"\"#!\"\"%\"xGF$" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "implicitplot(x^2 - 4* y ^2 = 36, x= -10..10, y = -10..10, grid=[40,40]);" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "e) Graph is a parabola. " }} {PARA 0 "" 0 "" {TEXT -1 13 "Vertex: (0,0)" }}{PARA 0 "" 0 "" {TEXT -1 13 "Focus: (1,0) " }}{PARA 0 "" 0 "" {TEXT -1 18 "Directrix: x = - 1" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "implicitplot(y^2 = 4*x, x = -10..10, y = -10..10, gri d=[40,40]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "f) \+ The graph is an ellispe. " }}{PARA 0 "" 0 "" {TEXT -1 14 "Center: (0 ,0)" }}{PARA 0 "" 0 "" {TEXT -1 12 "Foci: ( +- " }{XPPEDIT 18 0 "sqrt (2/3)" "6#-%%sqrtG6#*&\"\"#\"\"\"\"\"$!\"\"" }{TEXT -1 4 ", 0)" }} {PARA 0 "" 0 "" {TEXT -1 31 "Major axis vertices: (+- 1,0) " }}{PARA 0 "" 0 "" {TEXT -1 28 "Minor axis vertices: (0, +- " }{XPPEDIT 18 0 "s qrt(1/3)" "6#-%%sqrtG6#*&\"\"\"F'\"\"$!\"\"" }{TEXT -1 1 " " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "implicitplot(x^2 + 3* y^2 - \+ 1 = 0, x = -1..1, y = -1..1, grid=[40,40]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "g) Graph is a rotated hyperbola. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "implicitplot(x^2 + 3*x * y + y^2 = 16, x = -10..10, y = -10..10, grid=[40,40]);" }{TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "h) Graph is a parabola" }}{PARA 0 "" 0 "" {TEXT -1 15 "Vertex: (0,1) " }}{PARA 0 "" 0 "" {TEXT -1 14 "Focus: (0,0) " }}{PARA 0 "" 0 "" {TEXT -1 17 "Directrix: y = 2" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "implicitplot(x^2 + 4*y = 4, \+ x = -3..3, y = -3..3, grid=[40,40]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "completesquare(x^2 + y^2 - 2*x + 2*y + 2 \+ = 0,x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "c ompletesquare(%,y);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "Solution is one point: (1,-1)." }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 36 "Equations of Conics in Standard Form" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 263 9 "Example 2" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Write the equation of the circle that goe s through (0,0), (-4,0), and (0,6). " }}{PARA 0 "" 0 "" {TEXT -1 47 "D etermine the center and radius of the circle." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 56 "solve(\{f=0, 16 - 4*d + f =0, 36 + 6*e + f = 0 \},\{d,e,f\});" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "completesquare(x^2 + y^2 + 4*x - 6*y = 0, x);" }{TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Thus, center is: (-2,3) and rad ius is " }{XPPEDIT 18 0 "sqrt(13)" "6#-%%sqrtG6#\"#8" }{TEXT -1 2 ". \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "completesquare(2*x^2 + \+ 2*y^2 - 5*x -9*y + 11 = 0,x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 18 "Standard form: " }{XPPEDIT 18 0 "(x - 5/4)^2 + (y - \+ 9/4)^2 = 9/8" "6#/,&*$,&%\"xG\"\"\"*&\"\"&F(\"\"%!\"\"F,\"\"#F(*$,&%\" yGF(*&\"\"*F(F+F,F,F-F(*&F2F(\"\")F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 9 "center: " }{XPPEDIT 18 0 "(5/4,9/4)" "6$*&\"\"&\"\"\" \"\"%!\"\"*&\"\"*F%F&F'" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "radius: " }{XPPEDIT 18 0 "sqrt(9/8)" "6#-%%sqrtG6#*&\"\"*\"\"\"\"\") !\"\"" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "a) we u se: " }{XPPEDIT 18 0 "(y-k)^2 = 4*p*(x-h)" "6#/*$,&%\"yG\"\"\"%\"kG! \"\"\"\"#*(\"\"%F'%\"pGF',&%\"xGF'%\"hGF)F'" }{TEXT -1 3 ". " }} {PARA 0 "" 0 "" {TEXT -1 20 "p = |3 - (-5)| = 8. " }}{PARA 0 "" 0 "" {TEXT -1 16 "h = 3, k = -2. " }}{PARA 0 "" 0 "" {TEXT -1 14 "Equation is: " }{XPPEDIT 18 0 "(y + 2)^2 = 32*(x - 3)" "6#/*$,&%\"yG\"\"\"\" \"#F'F(*&\"#KF',&%\"xGF'\"\"$!\"\"F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "vertex is (3,-2). " }}{PARA 0 "" 0 "" {TEXT -1 23 "dir ectrix is: x = -5. " }}{PARA 0 "" 0 "" {TEXT -1 20 "focus is: (11,-2 ). " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "A:= implicitplot(y^2 = 20 * x, x = -10..10, y = -10..10):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "C:= textplot([-7,7,'directrix'], color = blue):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "E:= textplot([5.5,-2,'focus' ], color = blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "B:= pl ot([-5,t, t = -10..10], color = blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "DD:= pointplot([11,-2]):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "display(\{A,B,C,E,DD\});" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "completesquare(%,y);" }{TEXT -1 0 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 9 "Example 3" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 109 "Write the equati on of the ellipse in standard form and determine its center, vertices, foci, and covertices. " }}{PARA 0 "" 0 "" {TEXT -1 30 " Plot the elli pse with Foci. " }}{PARA 0 "" 0 "" {TEXT -1 28 " \+ " }{XPPEDIT 18 0 "4*x^2 + 5*y^2 - 24*x - 10*y + 17 = 0" "6#/,,* &\"\"%\"\"\"*$%\"xG\"\"#F'F'*&\"\"&F'*$%\"yGF*F'F'*&\"#CF'F)F'!\"\"*& \"#5F'F.F'F1\"# " 0 "" {MPLTEXT 1 0 20 "completesquare(%,x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Standard form: " }{XPPEDIT 18 0 "(x - 3)^2 / 6 + (y-1)^2 /(24/5) = 1" "6#/,&*&,&%\"xG\"\"\"\"\"$!\"\"\"\"#\"\"'F*F( *&,&%\"yGF(F(F*F+*&\"#CF(\"\"&F*F*F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 "Center: (3,1) " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "a^2 = 6" "6#/*$%\"aG\"\"#\"\"'" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b^2 = 24/5" "6#/*$%\"bG\"\"#*&\"#C\"\"\"\"\"&!\"\"" }{TEXT -1 7 ". Use " } {XPPEDIT 18 0 "a^2 - b^2 = c^2" "6#/,&*$%\"aG\"\"#\"\"\"*$%\"bGF'!\" \"*$%\"cGF'" }{TEXT -1 10 " to get: " }{XPPEDIT 18 0 "c^2 = 6/5" "6#/ *$%\"cG\"\"#*&\"\"'\"\"\"\"\"&!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 " " {TEXT -1 17 "Vertices: (3 +- " }{XPPEDIT 18 0 "sqrt(6)" "6#-%%sqrtG 6#\"\"'" }{TEXT -1 4 ",1) " }}{PARA 0 "" 0 "" {TEXT -1 21 "Covertices: (3, 1 +- " }{XPPEDIT 18 0 "sqrt(24/5)" "6#-%%sqrtG6#*&\"#C\"\"\"\"\"& !\"\"" }{TEXT -1 2 ") " }}{PARA 0 "" 0 "" {TEXT -1 14 "Foci: ( 3 +- \+ " }{XPPEDIT 18 0 "sqrt(6/5)" "6#-%%sqrtG6#*&\"\"'\"\"\"\"\"&!\"\"" } {TEXT -1 4 ",1)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "A:= imp licitplot(4 * x^2 + 5*y^2 - 24*x - 10*y + 17 = 0, x =-1..6 ,y = -3..4) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "C:= pointplot([3 -1.09 5445115, 1]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 9 "Example 4" }{TEXT -1 100 "\nWrite the equation of the \+ hyperbola in standard form and determine its center, vertices, and foc i. " }}{PARA 0 "" 0 "" {TEXT -1 55 "Plot the graph, state and plot the asymtotes and Foci. " }}{PARA 0 "" 0 "" {TEXT -1 29 " \+ " }{XPPEDIT 18 0 "3*x^2 - y^2 - 18*x + 10*y - 10 = 0 " "6#/,,*&\"\"$\"\"\"*$%\"xG\"\"#F'F'*$%\"yGF*!\"\"*&\"#=F'F)F'F-*&\"# 5F'F,F'F'F1F-\"\"!" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "completesquare(%,x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Standard form: " }{XPPEDIT 18 0 "(x - 3)^2 /4 \+ - (y-5)^2 / 12 = 1." "6#/,&*&,&%\"xG\"\"\"\"\"$!\"\"\"\"#\"\"%F*F( *&,&%\"yGF(\"\"&F*F+\"#7F*F*-%&FloatG6$F(\"\"!" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {XPPEDIT 18 0 "a^2 = 4" "6#/*$%\"aG\"\"#\"\"%" } {TEXT -1 6 " and " }{XPPEDIT 18 0 "b^2 = 12" "6#/*$%\"bG\"\"#\"#7" }{TEXT -1 8 " give " }{XPPEDIT 18 0 "c^2 = 16" "6#/*$%\"cG\"\"#\"#; " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "Vertices: (3 +-2,5) " }}{PARA 0 "" 0 "" {TEXT -1 18 "Foc i: (3 +- 4,5) " }}{PARA 0 "" 0 "" {TEXT -1 15 "Center: (3,5) " }} {PARA 0 "" 0 "" {TEXT -1 21 "asymptotes: y = +- " }{XPPEDIT 18 0 "sq rt(12)/2 * x" "6#*(-%%sqrtG6#\"#7\"\"\"\"\"#!\"\"%\"xGF(" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "f:= x -> 5 + 1.73205 0808 * (x-3) ;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "A:= implicitplot((x-3)^2/4-(y-5)^2/12 = 1, x = -10..12, y = -20. .20):\nB:= pointplot([7,5]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "E:= plot(f(x), x = -10..10, color= blue):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 21 "display(\{A,B,C,E,F\});" }{TEXT -1 0 "" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 265 10 "\nExample 5" }{TEXT -1 65 "\nUsin g the points (0,0), (-4,0), (0,6) and the general equation " } {XPPEDIT 18 0 "x^2 + y^2 + D*x + E*y + F = 0" "6#/,,*$%\"xG\"\"#\"\"\" *$%\"yGF'F(*&%\"DGF(F&F(F(*&%\"EGF(F*F(F(%\"FGF(\"\"!" }{TEXT -1 1 " \+ " }}{PARA 0 "" 0 "" {TEXT -1 11 "we obtain: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 " F = 0 " }}{PARA 0 "" 0 " " {TEXT -1 24 " 16 - 4D + F = 0 " }}{PARA 0 "" 0 "" {TEXT -1 23 " 36 + 6E + F = 0. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Eq uation of circle is: " }{XPPEDIT 18 0 "x^2 + y^2 + 4*x -6*y = 0" "6#/,**$%\"xG\"\"#\"\"\"*$%\"yGF'F(*&\"\"%F(F&F(F(*&\"\"'F(F*F(!\"\"\" \"!" }{TEXT -1 2 ". " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "com pletesquare(%,y);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 9 "Example 6" }{TEXT -1 72 "\nWrite t he equation of the parabola that has the given characteristics. " }} {PARA 0 "" 0 "" {TEXT -1 91 "Plot Each parabola with Focus and direc trix. Label the focus and directirx on the graph. " }}{PARA 0 "" 0 "" {TEXT -1 39 "a) vertex (3,-2) and directrix x = -5 " }}{PARA 0 "" 0 " " {TEXT -1 80 "b) passes through (5,10), vertex is at the origin, and the axis is the x-axis. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "A:= implicitplot((y+2)^2 = 32*(x-3), x = -10..12, y = -20..20):" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "C:= textplot([-7,7,'direct rix'], color = blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "E: = textplot([11.5,-5,'focus'], color = blue):" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 13 "c) We use: " }{XPPEDIT 18 0 "(y - k)^2 = 4*p*(x-h)" " 6#/*$,&%\"yG\"\"\"%\"kG!\"\"\"\"#*(\"\"%F'%\"pGF',&%\"xGF'%\"hGF)F'" } {TEXT -1 49 ". Vertex is (0,0) implies that h = 0 and k = 0." }} {PARA 0 "" 0 "" {TEXT -1 38 "Since (5,10) is on the graph we have: " } }{PARA 0 "" 0 "" {TEXT -1 31 "100 = 4 p 5 = 20 p OR p = 5. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 " Thus equ ation is: " }{XPPEDIT 18 0 "y^2 = 20* x" "6#/*$%\"yG\"\"#*&\"#?\"\"\" %\"xGF)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 18 "Vertex is: ( 0,0) " }}{PARA 0 "" 0 "" {TEXT -1 17 "Focus is: (5,0) " }}{PARA 0 "" 0 "" {TEXT -1 25 "Directrix is: x = -5. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "B:= plot([-5,t,t = -10..10], color = blue):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "DD:= pointplot([5,0] ):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "display(\{A,B,C,DD,E\});" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT 268 9 "Example 7" }{TEXT -1 97 "\nWrite the equation of the pa rabola in standard form and determine its vertex, focus, directrix. " }}{PARA 0 "" 0 "" {TEXT -1 27 " " }{XPPEDIT 18 0 "x^2 - 4*x + 8*y + 36 = 0" "6#/,**$%\"xG\"\"#\"\"\"*&\"\"%F(F&F(! \"\"*&\"\")F(%\"yGF(F(\"#OF(\"\"!" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 42 "completesquare(x^2 -4*x + 8*y + 36 = 0,x);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Standard form: \+ " }{XPPEDIT 18 0 "(x - 2)^2 = -8*(y + 4)" "6#/*$,&%\"xG\"\"\"\"\"#!\" \"F(,$*&\"\")F',&%\"yGF'\"\"%F'F'F)" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 15 "Vertex: (2,-4)" }}{PARA 0 "" 0 "" {TEXT -1 15 "Focus: \+ (2,-6) " }}{PARA 0 "" 0 "" {TEXT -1 18 "Directrix: y = -2" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "A:= implicitplot(x^2-4*x+8* y+36 = 0, x = -10..12, y = -20..20):B:= plot([t,-2,t = -10..10], color = blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "DD:= pointplot ([2,-6] ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "display(\{A,B ,C,DD,E\});" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "completesquare(4*x^2 + 5*y^2 - 24*x - 10*y + 17 = 0, y);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "(%)/24;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "evalf(sqrt(6/5)); " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "B:= poin tplot([3 +1.095445115 , 1] ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "display(\{A,B,C\});" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "completesquare(3*x^2 - y^2 - 18*x + 10*y - 10 = 0,y); " }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "(%)/12;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "evalf(sqrt (12)/2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "g:= x -> 5 -1. 732050808 * (x - 3) ;" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "A:=implicitplot(3*x^2 - y^2 - 18*x + 10*y - 10 = 0, x = -6..10, y = -10..10, grid=[40,40]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "C:= pointplot([-1,5] ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "F:= plot(g(x), x = -10..10, color= blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(A,C,F);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "0 0 0" 2 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }