{VERSION 6 0 "IBM INTEL NT" "6.0" } {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 "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 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 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 265 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 269 "" 1 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 270 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" 258 275 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 276 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 277 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 278 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 283 "" 0 1 0 0 0 0 1 0 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 0 128 1 2 1 2 2 2 2 1 1 1 1 } 1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 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 "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple P lot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 14 "The Parabola " }}{PARA 0 "" 0 " " {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 " " {TEXT -1 18 "Version: 23.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "Th e geometrical description of a parabola " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 122 "A parabola is the \+ collection of points in a plane which are at an equal distance from a \+ fixed point F and a fixed line L. 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Vrmsd$\\F_o7$$\"3kw=^gJy.hF_o$\"3IAxeLr;T\\F_o7$$\"3CK6Pi,#o6'F_o$\"3- !yb0@Tk%\\F_o7$$\"3\\CF$pf$[IhF_o$\"3Q@`O!oi>&\\F_o7$$\"3\"[U[(=vlVhF_ o$\"3UjT.70Gd\\F_o7$$\"371e7!QWr:'F_o$\"3W+q)Hw=F'\\F_o7$$\"3#o)fgnT`q hF_o$\"3)\\BwC)>6o\\F_o7$$\"3%4JTO\"3&G='F_o$\"3mK(o5znI(\\F_o7$$\"3M* yNC=#)p>'F_o$\"3[[%ypqZ(y\\F_o7$$\"3]r1(H6N'4iF_o$\"3_m)y9*y#Q)\\F_o7$ $\"3yqv$**)*RJA'F_o$\"3un:uFWC*)\\F_o7$$\"3!**p@*>'zgB'F_o$\"3gE%zSuGW *\\F_oF^_lF4F__lFa_lFd_lFg`lF[alFbal70F'F " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 43 "In general, the dis tance between two points" }{XPPEDIT 18 0 "``(x[1],y[1])" "6#-%!G6$&%\" xG6#\"\"\"&%\"yG6#F)" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(x[2],y[2] )" "6#-%!G6$&%\"xG6#\"\"#&%\"yG6#F)" }{TEXT -1 29 " in the coordinate \+ plane is: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "d=sqrt((x[2]-x[1])^2+(y[2]-y[1]^2))" "6#/%\"d G-%%sqrtG6#,&*$,&&%\"xG6#\"\"#\"\"\"&F,6#F/!\"\"F.F/,&&%\"yG6#F.F/*$&F 56#F/F.F2F/" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 241 173 173 {PLOTDATA 2 "6.-%'CURVESG6%7$7$$\"\"!F)F(7$$\"\" &F)$\"\"$F)-%&COLORG6&%$RGBGF($\"\")!\"\"F(-%*THICKNESSG6#\"\"#-F$6$7% F'7$F+F(F*-%'COLOURG6&F2F)F)F)-F$6&F&-%'SYMBOLG6#%'CIRCLEGF>-%&STYLEG6 #%&POINTG-F$6&F&-FD6#%(DIAMONDGF>FG-F$6&F&-FD6#%&CROSSGF>FG-%%TEXTG6%7 $$\"#aF5$\"#NF5Q((x~,y~)6\"-%%FONTG6$%*HELVETICAG\"#5-FV6%7$$!\"$F5$! \"%F5FgnFin-FV6%7$$\"$^&!\"#$\"$N$FjoQ&2~~~2Fhn-Fjn6$F\\oF4-FV6%7$$!#> Fjo$!#bFjoQ&1~~~1FhnF^p-%*AXESSTYLEG6#%%NONEG-%+AXESLABELSG6%Q!FhnF_q- Fjn6#%(DEFAULTG-%%VIEWG6$FbqFbq" 1 2 0 1 10 0 2 9 1 1 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 82 "This follows from the fact that the horizontal \+ distance between the two points is " }{XPPEDIT 18 0 "abs(x[2]-x[1])" " 6#-%$absG6#,&&%\"xG6#\"\"#\"\"\"&F(6#F+!\"\"" }{TEXT -1 46 ", while th e vertical distance between them is " }{XPPEDIT 18 0 "abs(y[2]-y[1])" "6#-%$absG6#,&&%\"yG6#\"\"#\"\"\"&F(6#F+!\"\"" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 229 "These distances can always be pictured a s the lengths of the two shorter sides of a right-angled triangle whos e hypotenuse is the line segment joining the two points, and with the \+ other two sides parallel to the coordinate axes. " }}{PARA 0 "" 0 "" {TEXT -1 62 "Applying Pythagoras' theorem then gives the distance form ula. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 27 "The equation of a parabola " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{TEXT 264 1 " a" }{TEXT -1 96 " be a positive real number. We shall derive the equa tion of the parabola with its focus on the " }{TEXT 268 1 "x" }{TEXT -1 18 " axis at the point" }{XPPEDIT 18 0 " ``(a,0)" "6#-%!G6$%\"aG\" \"!" }{TEXT -1 21 " and having the line " }{XPPEDIT 18 0 "x=-a" "6#/% \"xG,$%\"aG!\"\"" }{TEXT -1 15 " as directrix. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 379 330 330 {PLOTDATA 2 "67-%'CURVESG6%7S7$$\"3+++++++]i!#<$!\"&\"\"!7$$\"3Hpu 2W!\\pr&F*$!3YLLLe%G?y%F*7$$\"3!)Q;#3\\rCF&F*$!3OmmT&esBf%F*7$$\"3k()* e![d4%z%F*$!3ALL$3s%3zVF*7$$\"39Gg5Q:_NVF*$!3_LL$e/$QkTF*7$$\"3+#Q2()> 6?!RF*$!3ommT5=q]RF*7$$\"3\\],r*e'[?NF*$!3ILL3_>f_PF*7$$\"3!*\\?3*H>h9 $F*$!3K++vo1YZNF*7$$\"3sC011#z5y#F*$!3;LL3-OJNLF*7$$\"3^!)y*Q&[gRCF*$! 3p***\\P*o%Q7$F*7$$\"3\">*z)os%o6@F*$!3Kmmm\"RFj!HF*7$$\"3*=50v,[C%=F* $!33LL$e4OZr#F*7$$\"3?,%H&>BJh:F*$!3u*****\\n\\!*\\#F*7$$\"3()4k:FgU-8 F*$!3%)*****\\ixCG#F*7$$\"3Q^E@S38v5F*$!3#******\\KqP2#F*7$$\"3]l&*e!f Rf())!#=$!39LL3-TC%)=F*7$$\"3Kh?#>g*pzoFap$!3[mmm\"4z)e;F*7$$\"3=+?L%z 0tQ&Fap$!3Mmmmm`'zY\"F*7$$\"3kf^bF!)e!)QFap$!3#****\\(=t)eC\"F*7$$\"3g ^MiD>j_FFap$!3!ommmh5$\\5F*7$$\"3gg3eFvOP$!3)f***\\iXg#G'Fap7$$\"3u'4S$)zeVG%F`r$!3ndmmT&Q(RTFap7$ $\"3i=8y0wIz6F`r$!3%\\mmTg=><#Fap7$$\"3Ko&z$>pw!4'!#B$!3vDMLLe*e$\\!#? 7$$\"3+*37W-?9;\"F`r$\"3!=nm\"zRQb@Fap7$$\"3[$z&y!QI1:%F`r$\"3_,+](=>Y 2%Fap7$$\"3kw+PSCxZ%*F`r$\"3summ\"zXu9'Fap7$$\"3^cSppwj<d*H=F*$\"3s++D1#=bq#F*7$$\"3M*)HY_=y=@F*$\"3 \"HLL$3s?6HF*7$$\"3(*>BZw,#QW#F*$\"3a***\\7`Wl7$F*7$$\"3)\\LM`#*)yyFF* $\"3enmmm*RRL$F*7$$\"3<%e*Q+`eVJF*$\"3%zmmTvJga$F*7$$\"3%H]<0&QdFNF*$ \"3]MLe9tOcPF*7$$\"3e22YN;#***QF*$\"31,++]Qk\\RF*7$$\"3Ao'=p7X'\\VF*$ \"3![LL3dg6<%F*7$$\"39\\d)p*)oEx%F*$\"3%ymmmw(GpVF*7$$\"3wG)**4\"*>`C& F*$\"3C++D\"oK0e%F*7$$\"3*>E)z&30'=dF*$\"35,+v=5s#y%F*7$F($\"\"&F--%'C OLOURG6&%$RGBG$\"#5!\"\"$F-F-Fa[l-%*THICKNESSG6#\"\"#-F$6%7$7$$F`[lF-F +7$Fj[lFhz-F[[l6&F][lF-F-F--%*LINESTYLEG6#\"\"\"-F$6&7#7$$Fa\\lF-Fa[l- F[[l6&F][lFa[l$\"*++++\"!\")Fa[l-%'SYMBOLG6$%'CIRCLEGF_[l-%&STYLEG6#%& POINTG-F$6&Fd\\lFg\\l-F]]l6$%(DIAMONDGF_[lF`]l-F$6&Fd\\lFg\\l-F]]l6$%& CROSSGF_[lF`]l-F$6&Fd\\lF\\\\l-F]]l6$F_]l\"#7F`]l-F$6$7%7$Fj[l$\"\"$F- 7$$\"3+++++++]AF*Fg^lFe\\l-%&COLORG6&F][lFa[l$\"\")F`[lFa[l-F$6&7$Ff^l Fi^l-F]]l6$F_]l\"\"(F\\\\lF`]l-F$6&Fc_lF\\]lF\\\\lF`]l-F$6&Fc_lF`^lF\\ \\lF`]l-%%TEXTG6&7$$Ff_lF-$!\"$F`[lQ\"x6\"-F]_l6&F][l$Fa\\l!\"#Ff`lFf` l-%%FONTG6$%*HELVETICAG\"\"*-F\\`l6&7$F``lFhzQ\"yFc`lFd`lFh`l-F\\`l6&7 $$\"#8F`[l$F,F`[lQ'F(a,0)Fc`lFd`l-Fi`l6$F[alF_[l-F\\`l6&7$$\"#JF`[l$\" #GF`[lQ'P(x,y)Fc`lFd`lFhal-F\\`l6&7$$!$&>Fg`lFg^lQ(M(-a,y)Fc`lFd`lFhal -F\\`l6&7$$F*F`[l$!#YF`[lQ'x~=~-aFc`lFd`lFhal-%*AXESTICKSG6$F-F--%(SCA LINGG6#%,CONSTRAINEDGFh`l-%+AXESLABELSG6%Q!Fc`lFicl-Fi`l6#%(DEFAULTG-% %VIEWG6$F\\dlF\\dl" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 43.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Cu rve 13" "Curve 14" "Curve 15" "Curve 16" }}}{PARA 0 "" 0 "" {TEXT -1 24 "If P is a general point " }{XPPEDIT 18 0 "``(x,y)" "6#-%!G6$%\"xG% \"yG" }{TEXT -1 169 " on the parabola, the distance FP between the foc us F and P, is equal to the horizontal distance from the point P to a \+ point M on the directrix at the same height as P. " }}{PARA 0 "" 0 "" {TEXT -1 35 "First note that the distance MP is " }{XPPEDIT 18 0 "x+a " "6#,&%\"xG\"\"\"%\"aGF%" }{TEXT -1 50 ". The distance formula is not needed to see this. " }}{PARA 0 "" 0 "" {TEXT -1 76 "However, we do n eed the distance formula to find the distance FP, which is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt((x-a)^2+(y-0)^2)=sqrt( (x-a)^2+y^2)" "6#/-%%sqrtG6#,&*$,&%\"xG\"\"\"%\"aG!\"\"\"\"#F+*$,&%\"y GF+\"\"!F-F.F+-F%6#,&*$,&F*F+F,F-F.F+*$F1F.F+" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Wherever \+ the point " }{XPPEDIT 18 0 "P(x,y)" "6#-%\"PG6$%\"xG%\"yG" }{TEXT -1 54 " is located, the distances MP and FP are equal. Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x+a=sqrt((x-a)^2+y^2)" "6#/ ,&%\"xG\"\"\"%\"aGF&-%%sqrtG6#,&*$,&F%F&F'!\"\"\"\"#F&*$%\"yGF/F&" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "Squaring both sides of \+ this equation gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x+a)^2 = (x-a)^2+y^2;" "6#/*$,&%\"xG\"\"\"%\"aGF'\"\"#,&*$,&F&F 'F(!\"\"F)F'*$%\"yGF)F'" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2+2 *a*x+a^2=x^2-2*a*x+a^2+y^2" "6#/,(*$%\"xG\"\"#\"\"\"*(F'F(%\"aGF(F&F(F (*$F*F'F(,**$F&F'F(*(F'F(F*F(F&F(!\"\"*$F*F'F(*$%\"yGF'F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 13 "which gives, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "4*a*x=y^2" "6#/*(\"\"%\"\"\"%\"aGF&% \"xGF&*$%\"yG\"\"#" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^2=4*a*x" "6#/*$ %\"yG\"\"#*(\"\"%\"\"\"%\"aGF)%\"xGF)" }{TEXT -1 2 ". " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{TEXT 265 6 "______" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "This equation is \+ satisfied by the coordinates of a point" }{XPPEDIT 18 0 "``(x,y)" "6#- %!G6$%\"xG%\"yG" }{TEXT -1 91 " exactly when the point lies on the par abola, that is, it is the equation of the parabola. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{TEXT 266 1 "a" } {TEXT -1 93 " be a positive real number. We now derive the equation of the parabola with its focus on the " }{TEXT 267 1 "y" }{TEXT -1 18 " \+ axis at the point" }{XPPEDIT 18 0 "``(0, a);" "6#-%!G6$\"\"!%\"aG" } {TEXT -1 21 " and having the line " }{XPPEDIT 18 0 "y = -a;" "6#/%\"yG ,$%\"aG!\"\"" }{TEXT -1 15 " as directrix. " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 363 374 374 {PLOTDATA 2 "67-%'CURVESG6%7S7$$ !\"&\"\"!$\"3+++++++]i!#<7$$!3YLLLe%G?y%F-$\"3Hpu2W!\\pr&F-7$$!3OmmT&e sBf%F-$\"3!)Q;#3\\rCF&F-7$$!3ALL$3s%3zVF-$\"3k()*e![d4%z%F-7$$!3_LL$e/ $QkTF-$\"39Gg5Q:_NVF-7$$!3ommT5=q]RF-$\"3+#Q2()>6?!RF-7$$!3ILL3_>f_PF- $\"3\\],r*e'[?NF-7$$!3K++vo1YZNF-$\"3!*\\?3*H>h9$F-7$$!3;LL3-OJNLF-$\" 3sC011#z5y#F-7$$!3p***\\P*o%Q7$F-$\"3^!)y*Q&[gRCF-7$$!3Kmmm\"RFj!HF-$ \"3\">*z)os%o6@F-7$$!33LL$e4OZr#F-$\"3*=50v,[C%=F-7$$!3u*****\\n\\!*\\ #F-$\"3?,%H&>BJh:F-7$$!3%)*****\\ixCG#F-$\"3()4k:FgU-8F-7$$!3#******\\ KqP2#F-$\"3Q^E@S38v5F-7$$!39LL3-TC%)=F-$\"3]l&*e!fRf())!#=7$$!3[mmm\"4 z)e;F-$\"3Kh?#>g*pzoFcp7$$!3Mmmmm`'zY\"F-$\"3=+?L%z0tQ&Fcp7$$!3#****\\ (=t)eC\"F-$\"3kf^bF!)e!)QFcp7$$!3!ommmh5$\\5F-$\"3g^MiD>j_FFcp7$$!3S$* **\\(=[jL)Fcp$\"3gg3eFvOP7$$ !3ndmmT&Q(RTFcp$\"3u'4S$)zeVG%Fbr7$$!3%\\mmTg=><#Fcp$\"3i=8y0wIz6Fbr7$ $!3vDMLLe*e$\\!#?$\"3Ko&z$>pw!4'!#B7$$\"3!=nm\"zRQb@Fcp$\"3+*37W-?9;\" Fbr7$$\"3_,+](=>Y2%Fcp$\"3[$z&y!QI1:%Fbr7$$\"3summ\"zXu9'Fcp$\"3kw+PSC xZ%*Fbr7$$\"3#4+++]y))G)Fcp$\"3^cSppwj<d*H=F-7$$\"3\"HLL$3s?6HF-$\"3M*)HY_=y=@F-7$$\"3a***\\7`Wl7$F-$ \"3(*>BZw,#QW#F-7$$\"3enmmm*RRL$F-$\"3)\\LM`#*)yyFF-7$$\"3%zmmTvJga$F- $\"3<%e*Q+`eVJF-7$$\"3]MLe9tOcPF-$\"3%H]<0&QdFNF-7$$\"31,++]Qk\\RF-$\" 3e22YN;#***QF-7$$\"3![LL3dg6<%F-$\"3Ao'=p7X'\\VF-7$$\"3%ymmmw(GpVF-$\" 39\\d)p*)oEx%F-7$$\"3C++D\"oK0e%F-$\"3wG)**4\"*>`C&F-7$$\"35,+v=5s#y%F -$\"3*>E)z&30'=dF-7$$\"\"&F*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$F*F*Fa[l-%* THICKNESSG6#\"\"#-F$6%7$7$F($F`[lF*7$FhzFj[l-F[[l6&F][lF*F*F*-%*LINEST YLEG6#\"\"\"-F$6&7#7$Fa[l$Fa\\lF*-F[[l6&F][lFa[l$\"*++++\"!\")Fa[l-%'S YMBOLG6$%'CIRCLEGF_[l-%&STYLEG6#%&POINTG-F$6&Fd\\lFg\\l-F]]l6$%(DIAMON DGF_[lF`]l-F$6&Fd\\lFg\\l-F]]l6$%&CROSSGF_[lF`]l-F$6&Fd\\lF\\\\l-F]]l6 $F_]l\"#7F`]l-F$6$7%7$$\"\"$F*Fj[l7$Fg^l$\"3+++++++]AF-Fe\\l-%&COLORG6 &F][lFa[l$\"\")F`[lFa[l-F$6&7$Ff^lFi^l-F]]l6$F_]l\"\"(F\\\\lF`]l-F$6&F c_lF\\]lF\\\\lF`]l-F$6&Fc_lF`^lF\\\\lF`]l-%%TEXTG6&7$Fhz$!\"$F`[lQ\"x6 \"-F]_l6&F][l$Fa\\l!\"#Fe`lFe`l-%%FONTG6$%*HELVETICAG\"\"*-F\\`l6&7$F_ `l$Ff_lF*Q\"yFb`lFc`lFg`l-F\\`l6&7$$!\"(F`[l$\"#8F`[lQ'F(0,a)Fb`lFc`l- Fh`l6$Fj`lF_[l-F\\`l6&7$$\"#QF`[l$\"#AF`[lQ'P(x,y)Fb`lFc`lFial-F\\`l6& 7$Fg^l$!#8F`[lQ(M(x,-a)Fb`lFc`lFial-F\\`l6&7$$!#YF`[lFfblQ'y~=~-aFb`lF c`lFial-%*AXESTICKSG6$F*F*-%(SCALINGG6#%,CONSTRAINEDGFg`l-%+AXESLABELS G6%Q!Fb`lFicl-Fh`l6#%(DEFAULTG-%%VIEWG6$F\\dlF\\dl" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" " Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" }} {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 24 "If P is a general point \+ " }{XPPEDIT 18 0 "``(x,y)" "6#-%!G6$%\"xG%\"yG" }{TEXT -1 182 " on the parabola, the distance FP between the focus F and P, is equal to the \+ vertical distance from the point P to a point M on the directrix at th e same horzontal distance from the " }{TEXT 270 1 "y" }{TEXT -1 12 " a xis as P. " }}{PARA 0 "" 0 "" {TEXT -1 35 "First note that the distanc e MP is " }{XPPEDIT 18 0 "y+a;" "6#,&%\"yG\"\"\"%\"aGF%" }{TEXT -1 2 " . " }}{PARA 0 "" 0 "" {TEXT -1 52 "Using the distance formula, to the \+ distance FP, is: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " sqrt((x-0)^2+(y-a)^2) = sqrt(x^2+(y-a)^2);" "6#/-%%sqrtG6#,&*$,&%\"xG \"\"\"\"\"!!\"\"\"\"#F+*$,&%\"yGF+%\"aGF-F.F+-F%6#,&*$F*F.F+*$,&F1F+F2 F-F.F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 "Wherever the point " }{XPPEDIT 18 0 "P(x,y)" "6#-%\" PG6$%\"xG%\"yG" }{TEXT -1 54 " is located, the distances MP and FP are equal. Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y+a = sqrt(x^2+(y-a)^2);" "6#/,&%\"yG\"\"\"%\"aGF&-%%sqrtG6#,&*$%\"xG\"\" #F&*$,&F%F&F'!\"\"F.F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "Squaring both sides of this equation gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(y+a)^2 = x^2+(y-a)^2;" "6#/*$,&%\"yG\" \"\"%\"aGF'\"\"#,&*$%\"xGF)F'*$,&F&F'F(!\"\"F)F'" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^2+2*a*y+a^2 = x^2+y^2-2*a*y+a^2;" "6#/,(*$%\"yG \"\"#\"\"\"*(F'F(%\"aGF(F&F(F(*$F*F'F(,**$%\"xGF'F(*$F&F'F(*(F'F(F*F(F &F(!\"\"*$F*F'F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 13 "whic h gives, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "4*a*y = \+ x^2;" "6#/*(\"\"%\"\"\"%\"aGF&%\"yGF&*$%\"xG\"\"#" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x^2 = 4*a*y;" "6#/*$%\"xG\"\"#*(\"\"%\"\"\"%\"aGF)%\"yG F)" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 269 6 " ______" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 4 "Note" }{TEXT -1 94 ": It was not really necessary to derive this equation independently of the previous equation " } {XPPEDIT 18 0 "y^2=4*a*x" "6#/*$%\"yG\"\"#*(\"\"%\"\"\"%\"aGF)%\"xGF) " }{TEXT -1 178 ". Since the roles of the two coordinate axes have bee n interchanged in the second derivation, the new equation can be obtai ned from the previous equation simply by interchanging " }{TEXT 271 1 "x" }{TEXT -1 5 " and " }{TEXT 272 1 "y" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 19 "Also note that, if " }{TEXT 282 1 "a" }{TEXT -1 14 " is positive, " }{XPPEDIT 18 0 "y^2 = -4*a*x" "6#/*$%\"yG\"\"#,$*( \"\"%\"\"\"%\"aGF*%\"xGF*!\"\"" }{TEXT -1 44 " is the equation of a pa rabola with focus at" }{XPPEDIT 18 0 " ``(-a,0)" "6#-%!G6$,$%\"aG!\"\" \"\"!" }{TEXT -1 15 " and directrix " }{XPPEDIT 18 0 "x=a" "6#/%\"xG% \"aG" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 382 350 350 {PLOTDATA 2 "67-%'CURVESG6%7S7$$!3+++++++]i!#<$!\"&\"\"!7$ $!3Hpu2W!\\pr&F*$!3YLLLe%G?y%F*7$$!3!)Q;#3\\rCF&F*$!3OmmT&esBf%F*7$$!3 k()*e![d4%z%F*$!3ALL$3s%3zVF*7$$!39Gg5Q:_NVF*$!3_LL$e/$QkTF*7$$!3+#Q2( )>6?!RF*$!3ommT5=q]RF*7$$!3\\],r*e'[?NF*$!3ILL3_>f_PF*7$$!3!*\\?3*H>h9 $F*$!3K++vo1YZNF*7$$!3sC011#z5y#F*$!3;LL3-OJNLF*7$$!3^!)y*Q&[gRCF*$!3p ***\\P*o%Q7$F*7$$!3\">*z)os%o6@F*$!3Kmmm\"RFj!HF*7$$!3*=50v,[C%=F*$!33 LL$e4OZr#F*7$$!3?,%H&>BJh:F*$!3u*****\\n\\!*\\#F*7$$!3()4k:FgU-8F*$!3% )*****\\ixCG#F*7$$!3Q^E@S38v5F*$!3#******\\KqP2#F*7$$!3]l&*e!fRf())!#= $!39LL3-TC%)=F*7$$!3Kh?#>g*pzoFap$!3[mmm\"4z)e;F*7$$!3=+?L%z0tQ&Fap$!3 Mmmmm`'zY\"F*7$$!3kf^bF!)e!)QFap$!3#****\\(=t)eC\"F*7$$!3g^MiD>j_FFap$ !3!ommmh5$\\5F*7$$!3gg3eFvOP$ !3)f***\\iXg#G'Fap7$$!3u'4S$)zeVG%F`r$!3ndmmT&Q(RTFap7$$!3i=8y0wIz6F`r $!3%\\mmTg=><#Fap7$$!3Ko&z$>pw!4'!#B$!3vDMLLe*e$\\!#?7$$!3+*37W-?9;\"F `r$\"3!=nm\"zRQb@Fap7$$!3[$z&y!QI1:%F`r$\"3_,+](=>Y2%Fap7$$!3kw+PSCxZ% *F`r$\"3summ\"zXu9'Fap7$$!3^cSppwj<d*H= F*$\"3s++D1#=bq#F*7$$!3M*)HY_=y=@F*$\"3\"HLL$3s?6HF*7$$!3(*>BZw,#QW#F* $\"3a***\\7`Wl7$F*7$$!3)\\LM`#*)yyFF*$\"3enmmm*RRL$F*7$$!3<%e*Q+`eVJF* $\"3%zmmTvJga$F*7$$!3%H]<0&QdFNF*$\"3]MLe9tOcPF*7$$!3e22YN;#***QF*$\"3 1,++]Qk\\RF*7$$!3Ao'=p7X'\\VF*$\"3![LL3dg6<%F*7$$!39\\d)p*)oEx%F*$\"3% ymmmw(GpVF*7$$!3wG)**4\"*>`C&F*$\"3C++D\"oK0e%F*7$$!3*>E)z&30'=dF*$\"3 5,+v=5s#y%F*7$F($\"\"&F--%'COLOURG6&%$RGBG$\"#5!\"\"$F-F-Fa[l-%*THICKN ESSG6#\"\"#-F$6%7$7$$\"\"\"F-F+7$Fj[lFhz-F[[l6&F][lF-F-F--%*LINESTYLEG 6#F[\\l-F$6&7#7$$F`[lF-Fa[l-F[[l6&F][lFa[l$\"*++++\"!\")Fa[l-%'SYMBOLG 6$%'CIRCLEGF_[l-%&STYLEG6#%&POINTG-F$6&Fd\\lFg\\l-F]]l6$%(DIAMONDGF_[l F`]l-F$6&Fd\\lFg\\l-F]]l6$%&CROSSGF_[lF`]l-F$6&Fd\\lF]\\l-F]]l6$F_]l\" #7F`]l-F$6$7%7$Fj[l$\"\"$F-7$$!3+++++++]AF*Fg^lFe\\l-%&COLORG6&F][lFa[ l$\"\")F`[lFa[l-F$6&7$Ff^lFi^l-F]]l6$F_]l\"\"(F]\\lF`]l-F$6&Fc_lF\\]lF ]\\lF`]l-F$6&Fc_lF`^lF]\\lF`]l-%%TEXTG6&7$$\"#>F`[l$!\"$F`[lQ\"x6\"-F] _l6&F][l$F[\\l!\"#Fg`lFg`l-%%FONTG6$%*HELVETICAG\"\"*-F\\`l6&7$Fa`lFhz Q\"yFd`lFe`lFi`l-F\\`l6&7$$!#8F`[l$F,F`[lQ(F(-a,0)Fd`lFe`l-Fj`l6$F\\al F_[l-F\\`l6&7$$!#JF`[l$\"#GF`[lQ'P(x,y)Fd`lFe`lFial-F\\`l6&7$$\"$&=Fh` lFg^lQ'M(a,y)Fd`lFe`lFial-F\\`l6&7$$\"#6?!RF-7$$!3ILL3_>f_PF-$!3\\],r*e'[ ?NF-7$$!3K++vo1YZNF-$!3!*\\?3*H>h9$F-7$$!3;LL3-OJNLF-$!3sC011#z5y#F-7$ $!3p***\\P*o%Q7$F-$!3^!)y*Q&[gRCF-7$$!3Kmmm\"RFj!HF-$!3\">*z)os%o6@F-7 $$!33LL$e4OZr#F-$!3*=50v,[C%=F-7$$!3u*****\\n\\!*\\#F-$!3?,%H&>BJh:F-7 $$!3%)*****\\ixCG#F-$!3()4k:FgU-8F-7$$!3#******\\KqP2#F-$!3Q^E@S38v5F- 7$$!39LL3-TC%)=F-$!3]l&*e!fRf())!#=7$$!3[mmm\"4z)e;F-$!3Kh?#>g*pzoFcp7 $$!3Mmmmm`'zY\"F-$!3=+?L%z0tQ&Fcp7$$!3#****\\(=t)eC\"F-$!3kf^bF!)e!)QF cp7$$!3!ommmh5$\\5F-$!3g^MiD>j_FFcp7$$!3S$***\\(=[jL)Fcp$!3gg3eFvOP7$$!3ndmmT&Q(RTFcp$!3u'4S$)zeVG% Fbr7$$!3%\\mmTg=><#Fcp$!3i=8y0wIz6Fbr7$$!3vDMLLe*e$\\!#?$!3Ko&z$>pw!4' !#B7$$\"3!=nm\"zRQb@Fcp$!3+*37W-?9;\"Fbr7$$\"3_,+](=>Y2%Fcp$!3[$z&y!QI 1:%Fbr7$$\"3summ\"zXu9'Fcp$!3kw+PSCxZ%*Fbr7$$\"3#4+++]y))G)Fcp$!3^cSpp wj<d*H=F-7$$\"3\"HLL$3s?6HF-$!3M*)HY_=y=@ F-7$$\"3a***\\7`Wl7$F-$!3(*>BZw,#QW#F-7$$\"3enmmm*RRL$F-$!3)\\LM`#*)yy FF-7$$\"3%zmmTvJga$F-$!3<%e*Q+`eVJF-7$$\"3]MLe9tOcPF-$!3%H]<0&QdFNF-7$ $\"31,++]Qk\\RF-$!3e22YN;#***QF-7$$\"3![LL3dg6<%F-$!3Ao'=p7X'\\VF-7$$ \"3%ymmmw(GpVF-$!39\\d)p*)oEx%F-7$$\"3C++D\"oK0e%F-$!3wG)**4\"*>`C&F-7 $$\"35,+v=5s#y%F-$!3*>E)z&30'=dF-7$$\"\"&F*F+-%'COLOURG6&%$RGBG$\"#5! \"\"$F*F*Fa[l-%*THICKNESSG6#\"\"#-F$6%7$7$F($\"\"\"F*7$FhzFj[l-F[[l6&F ][lF*F*F*-%*LINESTYLEG6#F[\\l-F$6&7#7$Fa[l$F`[lF*-F[[l6&F][lFa[l$\"*++ ++\"!\")Fa[l-%'SYMBOLG6$%'CIRCLEGF_[l-%&STYLEG6#%&POINTG-F$6&Fd\\lFg\\ l-F]]l6$%(DIAMONDGF_[lF`]l-F$6&Fd\\lFg\\l-F]]l6$%&CROSSGF_[lF`]l-F$6&F d\\lF]\\l-F]]l6$F_]l\"#7F`]l-F$6$7%7$$\"\"$F*Fj[l7$Fg^l$!3+++++++]AF-F e\\l-%&COLORG6&F][lFa[l$\"\")F`[lFa[l-F$6&7$Ff^lFi^l-F]]l6$F_]l\"\"(F] \\lF`]l-F$6&Fc_lF\\]lF]\\lF`]l-F$6&Fc_lF`^lF]\\lF`]l-%%TEXTG6&7$Fhz$! \"$F`[lQ\"x6\"-F]_l6&F][l$F[\\l!\"#Fe`lFe`l-%%FONTG6$%*HELVETICAG\"\"* -F\\`l6&7$F_`l$\"#;F`[lQ\"yFb`lFc`lFg`l-F\\`l6&7$$!\"(F`[l$!#8F`[lQ(F( 0,-a)Fb`lFc`l-Fh`l6$Fj`lF_[l-F\\`l6&7$$\"#QF`[l$!#AF`[lQ'P(x,y)Fb`lFc` lFjal-F\\`l6&7$Fg^lF_alQ'M(x,a)Fb`lFc`lFjal-F\\`l6&7$$!#YF`[l$\"#:F`[l Q&y~=~aFb`lFc`lFjal-%*AXESTICKSG6$F*F*-%+AXESLABELSG6%Q!Fb`lFfcl-Fh`l6 #%(DEFAULTGFg`l-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$FiclFicl" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "C urve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "C urve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" }}{TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 8 "Summary \+ " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 256 "" 0 "" {TEXT -1 3 " \+ " }{XPPEDIT 18 0 "matrix([[equation, focus, directrix, vertex, axis, o pens], [___________, ________, ________, ________, ________, _________ _________], [y^2 = 4*a*x, ``(a,0), x = -a, ``(0,0), x*axis, [`to right `, ` <`]], [y^2 = -4*a*x, ``(-a,0), x = a, ``(0,0), x*axis, [`to left `, ` >`]], [x^2 = 4*a*y, ``(0,a), y = -a, ``(0,0), y*axis, [upwards, ` union`(``,``)]], [x^2 = -4*a*y, ``(0,-a), y = a, ``(0,0), y*axis, [dow nwards, `intersect`(``,``)]]]);" "6#-%'matrixG6#7(7(%)equationG%&focus G%*directrixG%'vertexG%%axisG%&opensG7(%,___________G%)________GF0F0F0 %3__________________G7(/*$%\"yG\"\"#*(\"\"%\"\"\"%\"aGF9%\"xGF9-%!G6$F :\"\"!/F;,$F:!\"\"-F=6$F?F?*&F;F9F,F97$%*to~right~G%#~G7(/ *$F;F6*(F8F9F:F9F5F9-F=6$F?F:/F5,$F:FB-F=6$F?F?*&F5F9F,F97$%(upwardsG- %&unionG6$F=F=7(/*$F;F6,$*(F8F9F:F9F5F9FB-F=6$F?,$F:FB/F5F:-F=6$F?F?*& F5F9F,F97$%*downwardsG-%*intersectG6$F=F=" }{TEXT -1 4 " " }{TEXT 274 1 "a" }{TEXT -1 10 " positive " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "matrix([[equation, `alterna tive form`,focus, directrix, vertex, axis, opens], [``, `of equation`, ``, ``, ``,``, ``],[___________, ___________, ________, ________, ____ ____, ________, __________________], [y^2 = 4*a*x, x=y^2/(4*a),``(a,0) , x = -a, ``(0,0), x*axis, [`to right `, ` <`]], [y^2 = -4*a*x, x=-y^2 /(4*a),``(-a,0), x = a, ``(0,0), x*axis, [`to left`, ` >`]], [x^2 = 4* a*y,y=x^2/(4*a), ``(0,a), y = -a, ``(0,0), y*axis, [upwards, `union`(` `,``)]], [x^2 = -4*a*y,y=-x^2/(4*a), ``(0,-a), y = a, ``(0,0), y*axis, [downwards, `intersect`(``,``)]]])" "6#-%'matrixG6#7)7)%)equationG%1a lternative~formG%&focusG%*directrixG%'vertexG%%axisG%&opensG7)%!G%,of~ equationGF0F0F0F0F07)%,___________GF3%)________GF4F4F4%3______________ ____G7)/*$%\"yG\"\"#*(\"\"%\"\"\"%\"aGF=%\"xGF=/F?*&F9F:*&FF=!\" \"-F06$F>\"\"!/F?,$F>FC-F06$FFFF*&F?F=F-F=7$%*to~right~G%#~F=F?F=FC/F?,$*&F9F:*&FF=FCFC-F06$,$F>FCFF/F?F>-F06$FFFF *&F?F=F-F=7$%(to~leftG%#~>G7)/*$F?F:*(FF=F9F=/F9*&F?F:*&FF=F C-F06$FFF>/F9,$F>FC-F06$FFFF*&F9F=F-F=7$%(upwardsG-%&unionG6$F0F07)/*$ F?F:,$*(FF=F9F=FC/F9,$*&F?F:*&FF=FCFC-F06$FF,$F>FC/F9F>-F06$ FFFF*&F9F=F-F=7$%*downwardsG-%*intersectG6$F0F0" }{TEXT -1 4 " " } {TEXT 281 1 "a" }{TEXT -1 10 " positive " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT 277 8 "Question" }{TEXT -1 2 ": \+ " }}{PARA 0 "" 0 "" {TEXT -1 64 "(a) Find the focus and directrix of t he parabola with equation " }{XPPEDIT 18 0 "y=x^2" "6#/%\"yG*$%\"xG\" \"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 38 "(b) Sketch the gr aph of the parabola " }{XPPEDIT 18 0 "y=x^2" "6#/%\"yG*$%\"xG\"\"#" } {TEXT -1 46 ", indicating clearly the focus and directrix. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 8 "Solution" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 27 "(a) Comparing the equat ion " }{XPPEDIT 18 0 "y=x^2" "6#/%\"yG*$%\"xG\"\"#" }{TEXT -1 30 " wit h the \"standard\" equation " }{XPPEDIT 18 0 "x^2=4*a*y" "6#/*$%\"xG\" \"#*(\"\"%\"\"\"%\"aGF)%\"yGF)" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "4*a=1" "6#/*&\"\"%\"\"\"%\"aGF&F&" }{TEXT -1 14 ", which gives \+ " }{XPPEDIT 18 0 "a=1/4" "6#/%\"aG*&\"\"\"F&\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 43 " It follows that the focus is th e point" }{XPPEDIT 18 0 "``(0,1/4)" "6#-%!G6$\"\"!*&\"\"\"F(\"\"%!\"\" " }{TEXT -1 34 ", while the directrix is the line " }{XPPEDIT 18 0 "y= -1/4" "6#/%\"yG,$*&\"\"\"F'\"\"%!\"\"F)" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 4 "(b) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 238 " p1 := plot(x^2,x=-1.5..1.5,thickness=2):\np2 := plot(-1/4,x=-1.5..1.5, color=blue,thickness=2):\np3 := plot([[[0,1/4]]$3],style=point,color=b lack,symbol=[circle,diamond,cross]):\nplots[display]([p1,p2,p3],scalin g=constrained,tickmarks=[3,3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 283 236 236 {PLOTDATA 2 "6+-%'CURVESG6%7S7$$!3++++++++:!#<$\"3+++++++]AF*7$$!3 &*****\\P&3YV\"F*$\"3i))y'el,\"e?F*7$$!3!***\\iv$FX7$$!3')*****\\FPm( \\FX$\"3UU>pc=pwCFX7$$!3()*******4'*QS%FX$\"3%4_ff3I%R>FX7$$!3?++Dc>mP PFX$\"3'y&)>**o6qR\"FX7$$!3'3+++&=$z9$FX$\"3bFWCK\\Z4**!#>7$$!3N***\\i X/4]#FX$\"3C06H*4BXD'Fgq7$$!3C***\\(o8y%)=FX$\"3)47()z!3S_NFgq7$$!33** **\\i:#>C\"FX$\"3#!#B7$$\"3i(***\\P>:mkFgq$\"3?(\\$)y3 76=%F\\s7$$\"3d***\\iv&QA7FX$\"3S$)G3PpA%\\\"Fgq7$$\"3j++]PPBW=FX$\"3J @K`y!)>,MFgq7$$\"3%*)*****\\Nm'[#FX$\"3'pf)*3h&\\$='Fgq7$$\"36****\\(y b^6$FX$\"3SRZR!e&>/(*Fgq7$$\"3')***\\PMaKs$FX$\"3;C`#3HiiQ\"FX7$$\"3a* ***\\7TW)R%FX$\"3z!fy51JY$>FX7$$\"3*y*****\\@80]FX$\"3-gj*QyM^]#FX7$$ \"3_+++D6!Hl&FX$\"3)pi-H6Hb>$FX7$$\"3j)**\\P4w)RiFX$\"3Mu_`m`g$*QFX7$$ \"3s,++vZf\")oFX$\"3OK2tkYjNZFX7$$\"3'z**\\P/-a[(FX$\"3IkwlvV7.cFX7$$ \"3R++v=Yb;\")FX$\"3)o>:z)e%ye'FX7$$\"3s)****\\i@Ot)FX$\"3Jin')oYhFwFX 7$$\"3g)**\\PfL'z$*FX$\"3+^.INEv(z)FX7$$\"3>+++!*>=+5F*$\"3Qg.78SO+5F* 7$$\"3-++DE&4Q1\"F*$\"3T\\-932pJ6F*7$$\"3=+]P%>5p7\"F*$\"3s+j='eE*p7F* 7$$\"39+++bJ*[=\"F*$\"3Faew)yrRS\"F*7$$\"33++Dr\"[8D\"F*$\"3[>2pXA(ec \"F*7$$\"3++++Ijy58F*$\"3%*o[\"H!3;= " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "" {TEXT 275 8 "Question" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 81 "(a) Find the equation o f the parabola that has its vertex at the origin, has the " }{TEXT 279 1 "x" }{TEXT -1 59 " axis as its line of symmetry, and passes thro ugh the point" }{XPPEDIT 18 0 "``(-5,-10);" "6#-%!G6$,$\"\"&!\"\",$\"# 5F(" }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 71 "(b) Find the foc us and directrix of the parabola obtained in part (a). " }}{PARA 0 "" 0 "" {TEXT -1 95 "(c) Sketch the graph of the parabola found in (a), i ndicating clearly the focus and directrix. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 276 8 "Solution" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 77 "(a) The parabola must open to the left, \+ and so has an equation of the form " }{XPPEDIT 18 0 "y^2 = -4*a*x;" " 6#/*$%\"yG\"\"#,$*(\"\"%\"\"\"%\"aGF*%\"xGF*!\"\"" }{TEXT -1 8 ", wher e " }{TEXT 280 1 "a" }{TEXT -1 14 " is positive. " }}{PARA 0 "" 0 "" {TEXT -1 50 " Since the parabola passes through the point" } {XPPEDIT 18 0 "``(-5,-10);" "6#-%!G6$,$\"\"&!\"\",$\"#5F(" }{TEXT -1 10 ", we have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(-1 0)^2=-4*a*``(-5)" "6#/*$,$\"#5!\"\"\"\"#,$*(\"\"%\"\"\"%\"aGF,-%!G6#,$ \"\"&F'F,F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 13 " He nce " }{XPPEDIT 18 0 "100=20*a" "6#/\"$+\"*&\"#?\"\"\"%\"aGF'" }{TEXT -1 14 ", which gives " }{XPPEDIT 18 0 "a=5" "6#/%\"aG\"\"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 47 " The required parabola has the equation " }{XPPEDIT 18 0 "y^2 = -20*x;" "6#/*$%\"yG\"\"#,$*&\"# ?\"\"\"%\"xGF*!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "(b) The focus is the point" } {XPPEDIT 18 0 "``(-5,0);" "6#-%!G6$,$\"\"&!\"\"\"\"!" }{TEXT -1 34 ", \+ while the directrix is the line " }{XPPEDIT 18 0 "x = 5;" "6#/%\"xG\" \"&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 4 "(c) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 290 "p1 := plot([-5*t^2,10*t,t=-1.5..1.5],thickness=2):\np2 := plot([[5,-16],[5, 16]],color=blue,thickness=2):\np3 := plot([[[-5,0]]$3],style=point,col or=black,symbol=[circle,diamond,cross]):\nplots[display]([p1,p2,p3],sc aling=constrained,tickmarks=[3,3],view=[-11..7,-16..16],\n labels=[` x`,`y`]);" }}{PARA 13 "" 1 "" {GLPLOT2D 283 236 236 {PLOTDATA 2 "6+-%' CURVESG6%7S7$$!3+++++++D6!#;$!#:\"\"!7$$!3KWR$z#30H5F*$!3&*****\\P&3YV \"F*7$$!3a]*yMo[/\\*!#<$!3++]ivF6$!3u*****\\(4J@iF67$$!3AAhIE\"pwf\"F6$ !3!****\\iIKFl&F67$$!3CrfMGfMQ7F6$!3')*****\\FPm(\\F67$$!3A0wzH/:(p*!# =$!3()*******4'*QS%F67$$!3c*G*f\\%e])pF\\q$!3?++Dc>mPPF67$$!3y8A7muta \\F\\q$!3'3+++&=$z9$F67$$!3[_bk\\:EFJF\\q$!3N***\\iX/4]#F67$$!3agN*RS+ ix\"F\\q$!3C***\\(o8y%)=F67$$!3<'>7q$e%=r(!#>$!33****\\i:#>C\"F67$$!3T qjS!p`F7#Ffr$!3O!***\\7ev:lF\\q7$$!3WA$)[X!Qj4\"!#A$!3uF++](o2[\"Ffr7$ $!3_[<%R/c04#Ffr$\"3i(***\\P>:mkF\\q7$$!3+!z(4_[F\\q$\"36****\\(yb^6$F67$$!3v?m7a9JJpF\\q$\"3' )***\\PMaKs$F67$$!3?aHR0`:t'*F\\q$\"3+++]7TW)R%F67$$!37!=[>RnDD\"F6$\" 3*y*****\\@80]F67$$!3U88XcXw(f\"F6$\"3_+++D6!Hl&F67$$!31PwE$o-o%>F6$\" 3j)**\\P4w)RiF67$$!3'fOlBL=+5F*7$$!31Z7qSNXecF6$\"3/++DE&4Q1\"F*7$$!3//:$4$Hj\\jF6$ \"3=+]P%>5p7\"F*7$$!3yr#HQ%*e)>qF6$\"39+++bJ*[=\"F*7$$!3$yf`%G7OHyF6$ \"35++Dr\"[8D\"F*7$$!3gXVd9S!3f)F6$\"3++++Ijy58F*7$$!3R\"p*zRedT%*F6$ \"33+]P/)fTP\"F*7$$!3z'oVa\"*[$H5F*$\"3;+]i0j\"[V\"F*7$F($\"#:F--%'COL OURG6&%$RGBG$\"#5!\"\"$F-F-Fa[l-%*THICKNESSG6#\"\"#-F$6%7$7$$\"\"&F-$F *F-7$Fj[l$\"#;F--F[[l6&F][lFa[lFa[l$\"*++++\"!\")Fb[l-F$6&7#7$$!\"&F-F a[l-%'SYMBOLG6#%'CIRCLEG-F[[l6&F][lF-F-F--%&STYLEG6#%&POINTG-F$6&Fg\\l -F\\]l6#%(DIAMONDGF_]lFa]l-F$6&Fg\\l-F\\]l6#%&CROSSGF_]lFa]l-%*AXESTIC KSG6$\"\"$Fb^l-%+AXESLABELSG6%%\"xG%\"yG-%%FONTG6#%(DEFAULTG-%(SCALING G6#%,CONSTRAINEDG-%%VIEWG6$;$!#6F-$\"\"(F-;F\\\\lF^\\l" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve \+ 3" "Curve 4" "Curve 5" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 115 "Find the equation of the parabola which has its focus at the point ``(-1,1) and has the line y =x as its directrix. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 "" {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}} {PARA 0 "" 0 "" {TEXT -1 35 "___________________________________" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "__ _________________________________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 23 "Code for pictures etc. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 " " 0 "" {TEXT -1 43 "Code for animation of parabola construction" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 726 "p1 := plot([t^2,2*t,t=-2.5..2.5],thickness=2):\np2 := plot([[-1,- 5],[-1,5]],linestyle=1,color=black):\np3 := plot([[[1,0]]$4],style=poi nt,symbol=[circle,diamond,cross,circle],\n color=[green$3,black],sym bolsize=[10$3,12]):\ntt := 1.5:\np4 := plot([[-1,2*tt],[tt^2,2*tt],[1, 0]],color=COLOR(RGB,0,.8,0)):\np5 := plot([[[-1,2*tt],[tt^2,2*tt]]$3], style=point,symbol=[circle,diamond,cross],\n color=black,symbolsize =[10$3]):\nt1 := plots[textplot]([[7,-.3,`x`],[-.3,5,`y`]],font=[HELVE TICA,9],\n color=COLOR(RGB,.01,.01,.01)):\nt2 := plots[textplot]([[- 1.2,4.8,`L`],[.9,.3,`F`]],font=[TIMES,BOLD,11],\n color=COLOR(RGB,.0 1,.01,.01)):\nplots[display]([p1,p2,p3,p4,p5,t1,t2],font=[HELVETICA,9] ,\n scaling=constrained,tickmarks=[4,4]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1185 "numframes := 40:\np1 := plot([[-1,-5],[-1,5]],li nestyle=1,color=black):\np2 := plot([[[1,0]]$4],style=point,symbol=[ci rcle,diamond,cross,circle],\n color=[green$3,black],symbolsize=[10$3 ,12]):\nt1 := plots[textplot]([[7,-.3,`x`],[-.3,5,`y`]],font=[HELVETIC A,9],\n color=COLOR(RGB,.01,.01,.01)):\nfrms := NULL:\nnn := numfr ames-1:\ndd := evalf(5/nn):\nfor i from 0 to nn do\n tt := -2.5+dd*i ;\n p3 := plot([[-1,2*tt],[tt^2,2*tt],[1,0]],color=COLOR(RGB,0,.8,0) ):\n p4 := plot([[[-1,2*tt],[tt^2,2*tt]]$3],style=point,symbol=[circ le,diamond,cross],\n color=black,symbolsize=[10$3]):\n if i= 0 then\n p5 := plot([t^2,2*t,t=-2.5..2.5],color=black,linestyle=2 ,thickness=1):\n frms := frms,plots[display]([p||(1..5),t1]); \+ \n elif i=nn then\n p5 := plot([t^2,2*t,t=-2.5..2.5],color=red, thickness=3):\n frms := frms,plots[display]([p||(1..5),t1]);\n \+ else \n p5 := plot([t^2,2*t,t=-2.5..tt],color=red,thickness= 3):\n p6 := plot([t^2,2*t,t=tt..2.5],color=black,linestyle=2,thic kness=1):\n frms := frms,plots[display]([p||(1..6),t1]);\n end \+ if: \nend do:\nplots[display]([frms],insequence=true,font=[HELVETICA ,9],\n scaling=constrained,tickmarks=[4,4]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 26 "General parabola picture. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 762 "p1 : = plot([t^2,2*t,t=-2.5..2.5],thickness=2):\np2 := plot([[-1,-5],[-1,5] ],linestyle=1,color=black):\np3 := plot([[[1,0]]$4],style=point,symbol =[circle,diamond,cross,circle],\n color=[green$3,black],symbolsize=[ 10$3,12]):\ntt := 1.5:\np4 := plot([[-1,2*tt],[tt^2,2*tt],[1,0]],color =COLOR(RGB,0,.8,0)):\np5 := plot([[[-1,2*tt],[tt^2,2*tt]]$3],style=poi nt,symbol=[circle$3],\n color=black,symbolsize=[7,10,12]):\nt1 := p lots[textplot]([[7,-.3,`x`],[-.3,5,`y`]],font=[HELVETICA,9],\n color =COLOR(RGB,.01,.01,.01)):\nt2 := plots[textplot]([[1.3,-.5,`F(a,0)`],[ 3.1,2.8,`P(x,y)`],[-1.95,3,`M(-a,y)`],\n [-1.7,-4.6,`x = -a`]],font=[ HELVETICA,10],color=COLOR(RGB,.01,.01,.01)):\nplots[display]([p||(1..5 ),t1,t2],font=[HELVETICA,9],\n scaling=constrained,tickmarks=[0,0]) ;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 29 "Picture fo r distance formula " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 408 "p1 := plot([[0,0],[5,3]],color=COLOR(RGB,0 ,.8,0),thickness=2):\np2 := plot([[0,0],[5,0],[5,3]],color=black):\np3 := plot([[[0,0],[5,3]]$3],style=point,symbol=[circle,diamond,cross],c olor=black):\nt1 := plots[textplot]([[5.4,3.5,`(x ,y )`],[-.3,-.4,`(x \+ ,y )`]],font=[HELVETICA,10]):\nt2 := plots[textplot]([[5.51,3.35,`2 \+ 2`],[-.19,-.55,`1 1`]],font=[HELVETICA,8]):\nplots[display]([p1,p2,p 3,t1,t2],axes=none);\n" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 33 "Other general parabola pictures. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 766 "p1 := plot( [2*t,t^2,t=-2.5..2.5],thickness=2):\np2 := plot([[-5,-1],[5,-1]],lines tyle=1,color=black):\np3 := plot([[[0,1]]$4],style=point,symbol=[circl e,diamond,cross,circle],\n color=[green$3,black],symbolsize=[10$3,12 ]):\ntt := 1.5:\np4 := plot([[2*tt,-1],[2*tt,tt^2],[0,1]],color=COLOR( RGB,0,.8,0)):\np5 := plot([[[2*tt,-1],[2*tt,tt^2]]$3],style=point,symb ol=[circle$3],\n color=black,symbolsize=[7,10,12]):\nt1 := plots[te xtplot]([[5,-.3,`x`],[-.3,7,`y`]],font=[HELVETICA,9],\n color=C OLOR(RGB,.01,.01,.01)):\nt2 := plots[textplot]([[-.7,1.3,`F(0,a)`],[3. 8,2.2,`P(x,y)`],[3,-1.3,`M(x,-a)`],\n [-4.6,-1.3,`y = -a`]],font=[HEL VETICA,10],color=COLOR(RGB,.01,.01,.01)):\nplots[display]([p||(1..5),t 1,t2],font=[HELVETICA,9],\n scaling=constrained,tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 764 "p1 := plot([-t^2,2*t,t=-2.5..2.5],thickness=2):\np2 := plot([ [1,-5],[1,5]],linestyle=1,color=black):\np3 := plot([[[-1,0]]$4],style =point,symbol=[circle,diamond,cross,circle],\n color=[green$3,black] ,symbolsize=[10$3,12]):\ntt := 1.5:\np4 := plot([[1,2*tt],[-tt^2,2*tt] ,[-1,0]],color=COLOR(RGB,0,.8,0)):\np5 := plot([[[1,2*tt],[-tt^2,2*tt] ]$3],style=point,symbol=[circle$3],\n color=black,symbolsize=[7,10, 12]):\nt1 := plots[textplot]([[1.9,-.3,`x`],[-.3,5,`y`]],font=[HELVETI CA,9],\n color=COLOR(RGB,.01,.01,.01)):\nt2 := plots[textplot]([[-1. 3,-.5,`F(-a,0)`],[-3.1,2.8,`P(x,y)`],[1.85,3,`M(a,y)`],\n [1.7,-4.6,` x = a`]],font=[HELVETICA,10],color=COLOR(RGB,.01,.01,.01)):\nplots[dis play]([p||(1..5),t1,t2],font=[HELVETICA,9],\n scaling=constrained,t ickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 768 "p1 := plot([2*t,-t^2,t=-2.5..2.5],thickness= 2):\np2 := plot([[-5,1],[5,1]],linestyle=1,color=black):\np3 := plot([ [[0,-1]]$4],style=point,symbol=[circle,diamond,cross,circle],\n colo r=[green$3,black],symbolsize=[10$3,12]):\ntt := 1.5:\np4 := plot([[2*t t,1],[2*tt,-tt^2],[0,-1]],color=COLOR(RGB,0,.8,0)):\np5 := plot([[[2*t t,1],[2*tt,-tt^2]]$3],style=point,symbol=[circle$3],\n color=black, symbolsize=[7,10,12]):\nt1 := plots[textplot]([[5,-.3,`x`],[-.3,1.6,`y `]],font=[HELVETICA,9],\n color=COLOR(RGB,.01,.01,.01)):\nt2 := plots[textplot]([[-.7,-1.3,`F(0,-a)`],[3.8,-2.2,`P(x,y)`],[3,1.6,`M(x ,a)`],\n [-4.6,1.5,`y = a`]],font=[HELVETICA,10],color=COLOR(RGB,.01, .01,.01)):\nplots[display]([p||(1..5),t1,t2],font=[HELVETICA,9],\n \+ scaling=constrained,tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }