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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 "Time s" 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 Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -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 50 "Translations of standard ellipses and hyperbolas " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nana imo, 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 42 "A circle is a limiting case of an ellipse " }}{PARA 0 "" 0 "" {TEXT -1 66 "Consider a circle with its c entre at the origin and having radius " }{TEXT 266 1 "a" }{TEXT -1 2 " . " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 353 330 330 {PLOTDATA 2 "64-%'CURVESG6%7S7$$\"\"\"\"\"!$F*F*7$$\"3w\"4hRPij!**!#=$ \"3Ikwb#=y_O\"F/7$$\"3E8J#))4-Qn*F/$\"3%[#\\ff*)GLDF/7$$\"3-N5')yke[#* F/$\"3gLj&[K5J!QF/7$$\"3?goz=42`')F/$\"3j9NXR4U7]F/7$$\"3Sb](G._U!zF/$ \"3t_H-qceDhF/7$$\"3_\\$R'oTd#3(F/$\"3!GO#*3qU&fqF/7$$\"3?H$>jubk6'F/$ \"3!\\@@&\\!>8\"zF/7$$\"3VGuIc:x5]F/$\"3-$>p(Qh-a')F/7$$\"37C3nW'R,#QF /$\"3aK+16bcT#*F/7$$\"3$oY@iF'QDDF/$\"3s&Q\"[M\"oen*F/7$$\"3OB^hAo8X8F /$\"33)fVPO<\"4**F/7$$!3+qB/u(p5(f!#@$\"2%HhJ<#)******!#<7$$!33)\\T#fB [i8F/$\"3ap%>wGZn!**F/7$$!3)e:d.:#=YEF/$\"3PA>=\\D`V'*F/7$$!3%*yV1J*3J x$F/$\"3IzF#e`m3E*F/7$$!3V(\\AOF:B/&F/$\"3vU'yI2&oN')F/7$$!3[igNw%*[Rg F/$\"3!G;)RQ+BqzF/7$$!3/XUwYjJ*3(F/$\"3AfvOg?x_qF/7$$!3&=e_b?/U!zF/$\" 3u3HvXQF/7$$!3!p!RS4Nij'*F/$\"3o$p9m#Q%=d#F/7$$!3#p(pT>+.2**F/$\"31V ?00]Ug8F/7$$!3/gKG4>&*****F/$\"3vCA\\O%485$!#?7$$!3__Is=!Q%3**F/$!3q#4 *>c=8]8F/7$$!3)>b`sYlSn*F/$!3UNbFGIGKDF/7$$!37_J.8AEj#*F/$!3q4&=j`Bsw$ F/7$$!3%\\F)4Kh=u')F/$!3kENX')3zv\\F/7$$!3EB*z7xng%zF/$!37W(H!pUCrgF/7 $$!3XD84Pfc5rF/$!3Y?#o?\"yMJqF/7$$!3!Q%y6Ike[gF/$!3)4j2yeGL'zF/7$$!3g@ UD=%)o!*\\F/$!3I_Mh6Mil')F/7$$!3o+1s\"[\"ysPF/$!3#\\IN,%***4E*F/7$$!30 yCH\"el'3EF/$!3=V>(fp[Pl*F/7$$!3g67Cvnc\"H\"F/$!3;$RoI*>C;**F/7$$!3Ezy OHMQdIFas$!3W1O#>E`*****F/7$$\"3GIAj`Ks(G\"F/$!3lYZ3X=u;**F/7$$\"3EKRq X8/bDF/$!3U$[o%H'z!o'*F/7$$\"3@%)yb&Q)zNQF/$!36.2<#>x]B*F/7$$\"3O7w4w0 I.]F/$!3wHgb5wMe')F/7$$\"3@\\#)[*R]$4hF/$!3/a*[=H2o\"zF/7$$\"33/wY'Q+$ *4(F/$!3)4d)eg?sUqF/7$$\"3[f=s$Ry,!zF/$!3D1$H)zD(p_v\\F/7$$\"3!G$e@`4+D#*F/$!3i&>2ndo*fQF/7$$\"31mGAp))oa'*F /$!3%)f]nRQ=0EF/7$$\"3@zb'f`bp!**F/$!3/up91t'4O\"F/7$F($\"36YKhSr8/#)! #F-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-%*THICKNESSG6#\"\"#-F$6%7$7$F+F+7$$ \"3a+++SSDg')F/$\"3++++++++]F/-%&COLORG6&F\\[lF+$\"\")F_[lF+F`[l-F$6&7 #Fh[l-%'SYMBOLG6#%'CIRCLEG-Fjz6&F\\[lF*F*F*-%&STYLEG6#%&POINTG-F$6&Fd \\l-Ff\\l6#%(DIAMONDGFi\\lF[]l-F$6&Fd\\l-Ff\\l6#%&CROSSGFi\\lF[]l-F$6& 7#Fg[l-Fjz6&F\\[lF+$\"*++++\"!\")F+-Ff\\l6$Fh\\lF^[lF[]l-F$6&F[^lF\\^l -Ff\\l6$Fc]lF^[lF[]l-F$6&F[^lF\\^l-Ff\\l6$Fh]lF^[lF[]l-F$6&F[^lFi\\l-F f\\l6$Fh\\l\"#7F[]l-%%TEXTG6&7$$\"$:\"!\"#$!\"&Ff_lQ\"x6\"Fi\\l-%%FONT G6$%*HELVETICAGF^[l-Fa_l6&7$Fg_lFd_lQ\"yFj_lFi\\lF[`l-Fa_l6&7$$\"$2\"F f_l$\"#`Ff_lQ'P(x,y)Fj_lFi\\lF[`l-Fa_l6&7$$!#8Ff_l$\"#6Ff_lQ&(0,0)Fj_l Fi\\lF[`l-Fa_l6&7$$\"#UFf_l$\"#NFf_lQ\"aFj_lFi\\l-F\\`l6%%&TIMESG%'ITA LICGFaal-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%Q!Fj_lFfbl-F\\`l6#% (DEFAULTG-%*AXESTICKSG6$F*F*-%%VIEWG6$;$!$:\"Ff_lFd_lF`cl" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curv e 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" }}{TEXT -1 3 " " } }{PARA 0 "" 0 "" {TEXT -1 8 "A point " }{XPPEDIT 18 0 "P(x,y)" "6#-%\" PG6$%\"xG%\"yG" }{TEXT -1 61 " lies on the circle exactly when its dis tance from the origin" }{XPPEDIT 18 0 " ``(0,0)" "6#-%!G6$\"\"!F&" } {TEXT -1 4 " is " }{TEXT 283 1 "a" }{TEXT -1 8 " units. " }}{PARA 0 " " 0 "" {TEXT -1 39 "Using the distance formula this gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(x^2+y^2)=a" "6#/-%%sqr tG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF*F+%\"aG" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 9 "that is, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x^2+y^2=a^2" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF'F(*$%\"aG F'" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 36 "This is the equati on of the circle. " }}{PARA 0 "" 0 "" {TEXT -1 126 "We may regard this circle as being a \"limiting case\" of an ellipse with its centre at \+ the origin, with its foci either on the " }{TEXT 284 1 "x" }{TEXT -1 9 " axis or " }{TEXT 285 1 "y" }{TEXT -1 192 " axis, where we have gra dually brought the two foci closer and closer to the origin, but kept \+ the constant sum of distances from any point on the ellipse to the two foci fixed and equal to 2. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 "For example, consider the ellipse with its foci at the points" }{XPPEDIT 18 0 "``(-1/10,0)" "6#-%!G6$,$*&\"\"\"F(\"#5 !\"\"F*\"\"!" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(1/10,0)" "6#-%!G6 $*&\"\"\"F'\"#5!\"\"\"\"!" }{TEXT -1 121 ", and such that the sum of t he distances from the foci to any point on the ellipse is 2. The equat ion of this ellipse is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2/a^2+y^2/b^2=1" "6#/,&*&%\"xG\"\"#*$%\"aGF'!\"\"\"\"\"*&%\"yG F'*$%\"bGF'F*F+F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "a = 1" "6#/%\"aG\" \"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "b = sqrt(1-c^2)" "6#/%\"bG-% %sqrtG6#,&\"\"\"F)*$%\"cG\"\"#!\"\"" }{TEXT -1 6 " with " }{XPPEDIT 18 0 "c=1/10" "6#/%\"cG*&\"\"\"F&\"#5!\"\"" }{TEXT -1 9 ". Hence " } {XPPEDIT 18 0 "b = sqrt(1-1/100);" "6#/%\"bG-%%sqrtG6#,&\"\"\"F)*&F)F) \"$+\"!\"\"F," }{XPPEDIT 18 0 "``=sqrt(99/100)" "6#/%!G-%%sqrtG6#*&\"# **\"\"\"\"$+\"!\"\"" }{XPPEDIT 18 0 "``=sqrt(99)/10" "6#/%!G*&-%%sqrtG 6#\"#**\"\"\"\"#5!\"\"" }{TEXT -1 1 " " }{TEXT 282 1 "~" }{TEXT -1 29 " 0.0995, and the equation is " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "x^2+100*y^2/99=1" "6#/,&*$%\"xG\"\"#\"\"\"*(\"$+\"F(*$% \"yGF'F(\"#**!\"\"F(F(" }{TEXT -1 15 " ------- (i). " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 276 1 "x" }{TEXT -1 5 " and " }{TEXT 281 1 "y" }{TEXT -1 25 " intercepts are given by " }{XPPEDIT 18 0 "x=`` " "6#/%\"xG%!G" }{TEXT 277 1 "+" }{TEXT -1 8 " 1 and " }{XPPEDIT 18 0 " y=``" "6#/%\"yG%!G" }{TEXT 278 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "s qrt(99)/10" "6#*&-%%sqrtG6#\"#**\"\"\"\"#5!\"\"" }{TEXT -1 1 " " } {TEXT 280 1 "~" }{TEXT -1 2 " " }{TEXT 279 1 "+" }{TEXT -1 22 " 0.099 5 respectively. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 122 "The graph of this ellipse looks very similar to the circ le with its centre at the origin and radius 1, which has equation " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2+y^2=1" "6#/,&*$% \"xG\"\"#\"\"\"*$%\"yGF'F(F(" }{TEXT -1 16 " ------- (ii). " }}{PARA 0 "" 0 "" {TEXT -1 2 " ." }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 321 302 302 {PLOTDATA 2 "67-%'CURVESG6%7S7$$\"\"\"\"\"!$F*F* 7$$\"3w\"4hRPij!**!#=$\"3mo]lRYVe8F/7$$\"3E8J#))4-Qn*F/$\"33^x3*o!f?DF /7$$\"3-N5')yke[#*F/$\"3)4/dZ*p/%y$F/7$$\"3?goz=42`')F/$\"3k%)>rjeH() \\F/7$$\"3Sb](G._U!zF/$\"3al+%fy!)[4'F/7$$\"3_\\$R'oTd#3(F/$\"3T]HN)Hc T-(F/7$$\"3?H$>jubk6'F/$\"3]*QsV1j;(yF/7$$\"3VGuIc:x5]F/$\"3yl\\a(GZ1h )F/7$$\"37C3nW'R,#QF/$\"3G*F/7$$\"3$oY@iF'QDDF/$\"3B0w(eBnti* F/7$$\"3OB^hAo8X8F/$\"3YRMo)GZ%f)*F/7$$!3+qB/u(p5(f!#@$\"3?X>D'fs)\\** F/7$$!33)\\T#fB[i8F/$\"3u(>:F4*3d)*F/7$$!3)e:d.:#=YEF/$\"3'yfOZt$>&f*F /7$$!3%*yV1J*3Jx$F/$\"3)ePx)eeW9#*F/7$$!3V(\\AOF:B/&F/$\"3Cx;gd\")R#f) F/7$$!3[igNw%*[RgF/$\"3a*>1#e(y-$zF/7$$!3/XUwYjJ*3(F/$\"3RF.4'f>u,(F/7 $$!3&=e_b?/U!zF/$\"37h?)3>U\\4'F/7$$!3w!G5)RnIf')F/$\"3Ejt$=Ppl(\\F/7$ $!3/`k8ti$4B*F/$\"3&4*\\:TeZEQF/7$$!3!p!RS4Nij'*F/$\"3wEwS(H_*eDF/7$$! 3#p(pT>+.2**F/$\"3KWu+*y0ON\"F/7$$!3/gKG4>&*****F/$\"3G8$fv#Rw&3$!#?7$ $!3__Is=!Q%3**F/$!3$ze6_BkLM\"F/7$$!3)>b`sYlSn*F/$!3NtC1+)*e>DF/7$$!37 _J.8AEj#*F/$!3KIw+\"4S$[PF/7$$!3%\\F)4Kh=u')F/$!3m_.;@%\\3&\\F/7$$!3EB *z7xng%zF/$!3eVGku<\"3/'F/7$$!3XD84Pfc5rF/$!3!pJ&=QF5'*pF/7$$!3!Q%y6Ik e[gF/$!3HW:d,>TBzF/7$$!3g@UD=%)o!*\\F/$!3\\l%G$Gk=A')F/7$$!3o+1s\"[\"y sPF/$!3\"3Q#\\%fyX@*F/7$$!30yCH\"el'3EF/$!3>;sWs'e`g*F/7$$!3g67Cvnc\"H \"F/$!3=#*[N0i`m)*F/7$$!3EzyOHMQdIF`s$!3mf6?my#)\\**F/7$$\"3GIAj`Ks(G \"F/$!3KB:$=bLq')*F/7$$\"3EKRqX8/bDF/$!3)evO7x<'>'*F/7$$\"3@%)yb&Q)zNQ F/$!35%3Nfy&y)=*F/7$$\"3O7w4w0I.]F/$!3X=Ao(4Z\\h)F/7$$\"3@\\#)[*R]$4hF /$!3I-(RlzBr(yF/7$$\"33/wY'Q+$*4(F/$!3n\"QoD(*>u+(F/7$$\"3[f=s$Ry,!zF/ $!3oMfCPt5+hF/7$$\"3%*R,@;vLu')F/$!3)Qcr,$oe]\\F/7$$\"3!G$e@`4+D#*F/$! 3ZI%Q5C?1%QF/7$$\"31mGAp))oa'*F/$!3_1@b\">D@f#F/7$$\"3@zb'f`bp!**F/$!3 ;sRyr`9a8F/7$F($\"3s#GNjQ8I;)!#F-%'COLOURG6&%$RGBG$\"*++++\"!\")F+F+-% *THICKNESSG6#\"\"#-F$6&7$7$$!3/+++++++5F/F+7$$\"3/+++++++5F/F+-Fiz6&F[ [lF+F\\[lF+-%'SYMBOLG6$%'CIRCLEG\"#5-%&STYLEG6#%&POINTG-F$6&Fe[lF\\\\l -F_\\l6$%(DIAMONDGFb\\lFc\\l-F$6&Fe[lF\\\\l-F_\\l6$%&CROSSGFb\\lFc\\l- F$6&Fe[l-Fiz6&F[[lF*F*F*-F_\\l6$Fa\\l\"#7Fc\\l-F$6$7%Ff[l7$$\"3;+++'G) )\\n#F/$\"3!)*****H*GG(e*F/Fi[l-%&COLORG6&F[[lF+$\"\")!\"\"F+-F$6&7#F[ ^lF^\\lFc]lFc\\l-F$6&Fh^lFi\\lFc]lFc\\l-F$6&Fh^lF^]lFc]lFc\\l-%%TEXTG6 &7$$F/!\"#$\"#9Fb_lQ\"F6\"-Fa^l6&F[[l$F)Fb_lFi_lFi_l-%%FONTG6$%*HELVET ICAGFb\\l-F^_l6&7$$\"#LFb_l$\"$3\"Fb_lQ'P(x,y)Ff_lFg_lFj_l-F^_l6&7$$\" #AFb_lFc_lFe_lFg_lFj_l-F^_l6&7$$!#9Fb_l$F)Fe^lQ\"1Ff_lFg_l-F[`l6$F]`lF d^l-F^_l6&7$$\"#EFb_lF`alQ\"2Ff_lFg_lFbal-F^_l6&7$$\"$:\"Fb_l$!\"'Fb_l Q\"xFf_lFg_l-F[`l6$F]`l\"\"*-F^_l6&7$F_blF]blQ\"yFf_lFg_lFbblFbbl-%*AX ESTICKSG6$\"\"%F\\cl-%+AXESLABELSG6%Q!Ff_lF`cl-F[`l6#%(DEFAULTG-%(SCAL INGG6#%,CONSTRAINEDG-%%VIEWG6$FcclFccl" 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 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Indeed solving equation (ii) for " }{TEXT 267 1 "y" }{TEXT -1 8 " \+ gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y=``" "6#/ %\"yG%!G" }{TEXT 270 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt(1-x^2) " "6#-%%sqrtG6#,&\"\"\"F'*$%\"xG\"\"#!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 32 "while, solving equation (i) for " }{TEXT 268 1 "y" }{TEXT -1 7 " gves: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y=``" "6#/%\"yG%!G" }{TEXT 269 1 "+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "sqrt(99)/100*sqrt(1-x^2)" "6#*(-%%sqrtG6#\"#**\"\"\"\"$ +\"!\"\"-F%6#,&F(F(*$%\"xG\"\"#F*F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 273 1 "y " }{TEXT -1 1 " " }{TEXT 271 1 "~" }{TEXT -1 1 " " }{TEXT 272 1 "+" } {TEXT -1 8 " 0.0995 " }{XPPEDIT 18 0 "sqrt(1-x^2)" "6#-%%sqrtG6#,&\"\" \"F'*$%\"xG\"\"#!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 88 "Thus we can obtain a point on the ell ipse from a point on the circle by multiplying its " }{TEXT 274 1 "y" }{TEXT -1 68 " coordinate by 0.0995, whereby it moves just slightly cl oser to the " }{TEXT 275 1 "x" }{TEXT -1 7 " axis. " }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 66 "Translating a \+ circle so that its centre moves away from the origin" }}{PARA 0 "" 0 " " {TEXT -1 46 "Consider a circle with its centre at the point" } {XPPEDIT 18 0 "``(h,k)" "6#-%!G6$%\"hG%\"kG" }{TEXT -1 19 " and having radius " }{TEXT 286 1 "a" }{TEXT -1 2 ". 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CG5HEF/7$$\"3aTI'[D,Bg)F/$\"31^9zY$Qu)GF/7$$\"3i]Z'fG+vw)F/$\"33x)Rty% )o8$F/7$$\"3egowgmS'*))F/$\"3op/>%=juT$F/7$$\"3au'*ygm3s*)F/$\"3az!)*) e%ycp$F/7$Fa_l$\"3a+X$=++++%F/FhzF_[l-F$6&7$7$Fc_lFc_l7$$Fe^lF*Fc_lF[ \\lF`\\lFd\\l-F$6&Fe^mF[\\lFj\\lFd\\l-F$6&Fe^mF[\\lF_]lFd\\l-F$6&Fe^mF d]lFf]lFd\\l-F$6$7%Ff^m7$$\"3!******f['\\-oF/$\"3Q+++-;eahF/Fg^mFa^l-F $6&7#Fb_mF`\\lFd]lFd\\l-F$6&Fi_mFj\\lFd]lFd\\l-F$6&Fi_mF_]lFd]lFd\\l-F $6%7$7$F(Fc_lF`_lFd]l-%*LINESTYLEG6#F)-F$6%7$7$$\"\"'F*$\"3'******H-KR w\"F/7$Fi`m$\"3E+++xz1OiF/Fd]lFb`m-%%TEXTG6&7$$\"\"\"F*$\"#EF^[lQ*(x-h ,y-k)6\"-Fb^l6&F[[l$FeamFh[lF\\bmF\\bm-%%FONTG6$%*HELVETICAGF][l-Faam6 &7$$\"\"(F*$\"$b'Fh[lQ&(x,y)FiamFjamF]bm-Faam6&7$$\"##*F^[l$Fh[lF^[lQ \"xFiamFjamF]bm-Faam6&7$$!#:Fh[l$\"$t'Fh[lQ\"yFiamFjamF]bm-%+AXESLABEL SG6%Q!FiamF[dm-F^bm6#%(DEFAULTG-F^bm6$F`bmFb_l-%*AXESTICKSG6$F*F*-%(SC ALINGG6#%,CONSTRAINEDG-%%VIEWG6$F^dmF^dm" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17 " "Curve 18" "Curve 19" "Curve 20" "Curve 21" "Curve 22" "Curve 23" "C urve 24" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 95 "We can determ ine the equation of the ellipse obtained by translating the ellipse wi th equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2/a ^2+y^2/b^2=1" "6#/,&*&%\"xG\"\"#*$%\"aGF'!\"\"\"\"\"*&%\"yGF'*$%\"bGF' F*F+F+" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 53 "so that its cen tre moves from the origin to the point" }{XPPEDIT 18 0 "``(h,k)" "6#-% !G6$%\"hG%\"kG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 7 "A point" }{XPPEDIT 18 0 " ``(x,y)" "6#-%!G 6$%\"xG%\"yG" }{TEXT -1 50 " lies on the translated ellipse with its c entre at" }{XPPEDIT 18 0 " ``(h,k)" "6#-%!G6$%\"hG%\"kG" }{TEXT -1 23 " exactly when the point" }{XPPEDIT 18 0 "``(x-h,y-k)" "6#-%!G6$,&% \"xG\"\"\"%\"hG!\"\",&%\"yGF(%\"kGF*" }{TEXT -1 74 " lies on the ellip se with its centre at the origin, that is, exactly when " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x-h)^2/(a^2)+(y-k)^2/(b^2) = 1 ;" "6#/,&*&,&%\"xG\"\"\"%\"hG!\"\"\"\"#*$%\"aGF+F*F(*&,&%\"yGF(%\"kGF* F+*$%\"bGF+F*F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{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 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT 309 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 41 "Sketch the ellipse given by the equation " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x-1)^2/9+(y-2)^2/4 = 1;" "6#/,&*&,&%\"xG\"\"\"F(!\"\"\"\"#\"\"*F)F(*&,&%\"yGF(F*F)F*\"\"%F )F(F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 51 "Identify the fo ci of the ellipse and determine any " }{TEXT 307 1 "x" }{TEXT -1 5 " a nd " }{TEXT 308 1 "y" }{TEXT -1 13 " intercepts. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 310 8 "Solution" }{TEXT -1 2 ": \+ " }}{PARA 0 "" 0 "" {TEXT -1 27 "The ellipse with equation " } {XPPEDIT 18 0 "x^2/9+y^2/4" "6#,&*&%\"xG\"\"#\"\"*!\"\"\"\"\"*&%\"yGF& \"\"%F(F)" }{TEXT -1 5 " has " }{TEXT 314 1 "x" }{TEXT -1 12 " interce pts " }{XPPEDIT 18 0 "x=``" "6#/%\"xG%!G" }{TEXT 312 1 "+" }{TEXT -1 7 " 3 and " }{TEXT 313 1 "y" }{TEXT -1 13 " intercepts " }{XPPEDIT 18 0 "y=``" "6#/%\"yG%!G" }{TEXT 311 1 "+" }{TEXT -1 4 " 2. " }}{PARA 0 "" 0 "" {TEXT -1 8 "Letting " }{XPPEDIT 18 0 "c=sqrt(9-4)" "6#/%\"cG -%%sqrtG6#,&\"\"*\"\"\"\"\"%!\"\"" }{XPPEDIT 18 0 "``=5" "6#/%!G\"\"& " }{TEXT -1 17 ", the foci are at" }{XPPEDIT 18 0 " ``(-c,0)" "6#-%!G6 $,$%\"cG!\"\"\"\"!" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(c,0)" "6#-% !G6$%\"cG\"\"!" }{TEXT -1 13 ", that is, at" }{XPPEDIT 18 0 " ``(-sqrt (5),0)" "6#-%!G6$,$-%%sqrtG6#\"\"&!\"\"\"\"!" }{TEXT -1 4 " and" } {XPPEDIT 18 0 " ``(sqrt(5),0)" "6#-%!G6$-%%sqrtG6#\"\"&\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "The ellipse with equation " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x-1)^2/9+(y-2)^2/4 \+ = 1" "6#/,&*&,&%\"xG\"\"\"F(!\"\"\"\"#\"\"*F)F(*&,&%\"yGF(F*F)F*\"\"%F )F(F(" }{TEXT -1 12 " ------- (i)" }}{PARA 0 "" 0 "" {TEXT -1 82 "is o btained by translating the first ellipse so that the centre moves to t he point" }{XPPEDIT 18 0 "``(1,2)" "6#-%!G6$\"\"\"\"\"#" }{TEXT -1 2 " . " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 476 476 476 {PLOTDATA 2 "67-%'CURVESG6%7S7$$\"\"$\"\"!$F*F*7$$\"3UF$)=7(3>(H!#<$\" 3iG`6ljbIF!#=7$$\"3(Q$pkH19-HF/$\"3o\\)*=>zdm]F27$$\"3i5$eO%fduFF/$\"3 ?nEr\\1A1wF27$$\"3%z0RcF@ff#F/$\"3$Hq!*y=%[-5F/7$$\"3s;D')4cFrBF/$\"3b !f/S8<^A\"F/7$$\"3v/=f]AxC@F/$\"3cs%y,a3>T\"F/7$$\"3w)z&*QsO\\$=F/$\"3 )HC/*4QE#e\"F/7$$\"3UGA*oYJK]\"F/$\"3gQQvF_!3t\"F/7$$\"3IZ7S$*=/Y6F/$ \"3]1?@-JJ[=F/7$$\"3/,WmG)ehd(F2$\"39xi*oit^$>F/7$$\"30q`%yY5a.%F2$\"3 i>([FZB=)>F/7$$!3+6F@K4K\"z\"!#?$\"3!fAjMk*****>F/7$$!3A%\\Cx2Zu3%F2$ \"3#R*Q_d%\\8)>F/7$$!3hn92^kaQzF2$\"3Z%QO)4lqG>F/7$$!3j8$>$zE$>8\"F/$ \"3'ebkrIt@&=F/7$$!3B\\n3#e%p7:F/$\"3bGdh9q8F-#z(f#F/$\"3&)e')*=3G.+\"F/7$$!3-O4%>)3GpFF/$\"3 gMt*)Re]\"p(F27$$!31s6#G0(3**GF/$\"3P(QHKl(oV^F27$$!3'H4De+4@(HF/$\"36 '3/,,]3s#F27$$!3Cy\\ysb)***HF/$\"3^\\W)H()=E?'Fbo7$$!3m:ph09`sHF/$!3T& =)R7PE+FF27$$!3glgA!HF/$!3%32^l0mX1&F27$$!3kX*4Rmy*yFF/$!3S>qjsqWM vF27$$!3P#[H'ReD-EF/$!3E`q!Hx\"e^**F27$$!3?xRQJ.#QQ#F/$!3#)[f!Q&)[U@\" F/7$$!3`(RF6ypJ8#F/$!34WOTi&piS\"F/7$$!39``.Hfd9=F/$!3=E:c_=F/7$$ !3rMu(Qu'*f#yF2$!3k)Q%>R(\\2$>F/7$$!3aMOsD.quQF2$!3kyOh)R[K)>F/7$$!3iQ O5)G]@<*Fbo$!3G@ZQ_1****>F/7$$\"3%3p'*3wpJ'QF2$!3M\\p,p$[L)>F/7$$\"3!o z6r.C^m(F2$!3o'p$*e#fhL>F/7$$\"3Altm:&R2:\"F/$!3iSTVQa,Z=F/7$$\"3g$GHG ^!G$=F/$!3!3zp$e9O$e\"F/7$$\"3B\" GSf6!zH@F/$!3?9x67Wa39F/7$$\"31el6=N0qBF/$!3DheM5x;E7F/7$$\"3)>/j[D,Bg #F/$!3;R'f^%R0^**F27$$\"3u\\Z'fG+vw#F/$!3C\"R9M:P*>xF27$$\"3qfowgmS'*G F/$!3p>,NzwO5_F27$$\"3mt'*ygm3sHF/$!31[RH7Y$>s#F27$F($\"3A\\E7Gu#3k\"! #E-%'COLOURG6&%$RGBG$\"*++++\"!\")F+F\\[l-%*THICKNESSG6#\"\"\"-F$6&7$7 $$!3#)*****p(z1OAF/F+7$$\"3#)*****p(z1OAF/F+Fhz-%'SYMBOLG6$%'CIRCLEG\" #5-%&STYLEG6#%&POINTG-F$6&Fe[lFhz-F]\\l6$%(DIAMONDGF`\\lFa\\l-F$6&Fe[l Fhz-F]\\l6$%&CROSSGF`\\lFa\\l-F$6&Fe[l-Fiz6&F[[lF*F*F*-F]\\l6$F_\\l\"# 7Fa\\l-F$6%7S7$$\"\"%F*$\"\"#F*7$$\"3UF$)=7(3>(RF/$\"3kK:^Oc0tAF/7$$\" 3(Q$pkH19-RF/$\"3>&)*=>zdm]#F/7$$\"3i5$eO%fduPF/$\"3sm7(\\1A1w#F/7$$\" 3%z0RcF@ff$F/$\"3$Hq!*y=%[-IF/7$$\"3s;D')4cFrLF/$\"3b!f/S8<^A$F/7$$\"3 v/=f]AxCJF/$\"3cs%y,a3>T$F/7$$\"3w)z&*QsO\\$GF/$\"3)HC/*4QE#e$F/7$$\"3 UGA*oYJK]#F/$\"3gQQvF_!3t$F/7$$\"3_Z7S$*=/Y@F/$\"3]1?@-JJ[QF/7$$\"35Sk 'G)ehd([FZB=)RF/7$$\"3?tyn! z'3#)**F2$\"3oDKYV'*****RF/7$$\"3x0bFAHb7fF2$\"39%*Q_d%\\8)RF/7$$\"3QK &G*[NXh?F2$\"3q%QO)4lqGRF/7$$!3GOJ>$zE$>8F2$\"3kbX;2L<_QF/7$$!3H#\\n3# e%p7&F2$\"3bGdh9q8FPF/7$$!3L'=o!H%o%=\")F2$\"3cK'zw+YSf$F/7$$!3_t#HS! \\zE6F/$\"3i6N27Wb5MF/7$$!3Wudmh7Er8F/$\"3`TB'oZH^A$F/7$$!3M%3V>-#z(f \"F/$\"32f')*=3G.+$F/7$$!3-O4%>)3GpF/$\"3y3/,,]3sAF/7$$!3Cy\\ysb)***>F /$\"3Y%)H()=E?1?F/7$$!3m:ph09`s>F/$\"3Y\"=g(GO(*HA!>F/$ \"3-$*[M%RVN\\\"F/7$$!3kX*4Rmy*yY6v&yF27$$!3`(RF6ypJ8\"F/$ \"33fN'eP/t$fF27$$!3UJNN!Hfd9)F2$\"31QZQCGMtSF27$$!3OlEwa_1s\\F2$\"3U& 4tn<`(oEF27$$!3;.=;XWM=8F2$\"3;!RH(>,+y9F27$$\"3GlD7cK+u@F2$\"3LO6c!3E ]#p!#>7$$\"3YljFu'*HDhF2$\"3'o8K'Q,;v;Fhhl7$$\"38k*=r\\y#3**F2$\"3G`(4 (y_hZ$*!#B7$$\"3'*o'*3wpJ'Q\"F/$\"3%p10$)4j^m\"Fhhl7$$\"3oz6r.C^m^!G$GF/$\"3#>4-jT&QmTF27$$\"3B\"GSf6!zHJF/ $\"30eG#)yeb9fF27$$\"31el6=N0qLF/$\"3](QTl*GKQxF27$$\"3)>/j[D,Bg$F/$\" 3(f.%[0Y*[+\"F/7$$\"3u\\Z'fG+vw$F/$\"3)3ceYG1!G7F/7$$\"3qfowgmS'*QF/$ \"3#z)\\1KK'*y9F/7$$\"3mt'*ygm3sRF/$\"3>01xQl!ys\"F/7$Fj]l$\"3[F3k,+++ ?F/-Fiz6&F[[l$F`\\l!\"\"F+F+-F`[l6#F]^l-F$6&7$7$$!3/+++xz1O7F/F\\^l7$$ \"3#)*****p(z1OKF/F\\^l-Fiz6&F[[lF+F\\[lF+F\\\\lFa\\l-F$6&Ff]mF]^mFg\\ lFa\\l-F$6&Ff]mF]^mF\\]lFa\\l-F$6&Ff]mFa]lFc]lFa\\l-F$6&7#7$$Fb[lF*F\\ ^l-F]\\l6#F_\\lFa]lFa\\l-F$6&Fg^m-F]\\l6#Fi\\lFa]lFa\\l-F$6&Fg^m-F]\\l 6#F^]lFa]lFa\\l-F$6%7$7$$!\"#F*F\\^lFi]lFa]l-%*LINESTYLEG6#F)-F$6%7$7$ Fi^mF+7$Fi^mFj]lFa]lFj_m-%%TEXTG6%7$$\"$L\"Fi_m$\"$B#Fi_mQ&(1,2)6\"-%& COLORG6&F[[l$Fb[lFi_mF_amF_am-Fc`m6%7$$\"#XFa]m$Fi_mFa]mQ\"xF[amF\\am- Fc`m6%7$FeamFcamQ\"yF[amF\\am-%+AXESLABELSG6%Q!F[amF^bm-%%FONTG6#%(DEF AULTG-%(SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$;$!#NFa]mFcam;$!#DFa]mFcam" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur ve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Cur ve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15 " "Curve 16" "Curve 17" "Curve 18" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 38 "The foci of the translated ellipse are" }{XPPEDIT 18 0 " \+ ``(1-sqrt(5),2)" "6#-%!G6$,&\"\"\"F'-%%sqrtG6#\"\"&!\"\"\"\"#" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(1+sqrt(5),2)" "6#-%!G6$,&\"\"\"F'-%%sq rtG6#\"\"&F'\"\"#" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "To \+ find any " }{TEXT 315 1 "x" }{TEXT -1 42 " intercepts of the translate d ellipse put " }{XPPEDIT 18 0 "y=0" "6#/%\"yG\"\"!" }{TEXT -1 18 " in equation (i). " }}{PARA 0 "" 0 "" {TEXT -1 12 "This gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x-1)^2/9+(0-2)^2/4 = 1;" " 6#/,&*&,&%\"xG\"\"\"F(!\"\"\"\"#\"\"*F)F(*&,&\"\"!F(F*F)F*\"\"%F)F(F( " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x-1)^2/9+1 = 1;" "6#/,&*&,&%\" xG\"\"\"F(!\"\"\"\"#\"\"*F)F(F(F(F(" }{TEXT -1 2 ", " }}{PARA 0 "" 0 " " {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " (x-1)^2 = 0.;" "6#/*$,&%\"xG\"\"\"F'!\"\"\"\"#-%&FloatG6$\"\"!F-" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 27 "Hence there is is just one " }{TEXT 317 1 "x" }{TEXT -1 17 " intercept where " }{XPPEDIT 18 0 "x=1" "6#/%\"xG\"\"\"" }{TEXT -1 31 ". The ellipse just touches the " }{TEXT 316 1 "x" }{TEXT -1 18 " axis at the point" }{XPPEDIT 18 0 "``(1,0)" "6#-%!G6$\"\"\"\"\"!" } {TEXT -1 15 ", that is, the " }{TEXT 318 1 "x" }{TEXT -1 54 " axis is \+ a tangent line to the ellipse at this point. " }}{PARA 0 "" 0 "" {TEXT -1 12 "To find any " }{TEXT 319 1 "y" }{TEXT -1 42 " intercepts \+ of the translated ellipse put " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"! " }{TEXT -1 18 " in equation (i). " }}{PARA 0 "" 0 "" {TEXT -1 12 "Thi s gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(0-1)^2/ 9+(y-2)^2/4 = 1;" "6#/,&*&,&\"\"!\"\"\"F(!\"\"\"\"#\"\"*F)F(*&,&%\"yGF (F*F)F*\"\"%F)F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so t hat " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(y-2)^2/4 = 1 -1/9;" "6#/*&,&%\"yG\"\"\"\"\"#!\"\"F(\"\"%F),&F'F'*&F'F'\"\"*F)F)" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 16 "It follows that " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(y-2)^2=32/9" "6#/*$, &%\"yG\"\"\"\"\"#!\"\"F(*&\"#KF'\"\"*F)" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 9 "and that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y-2=``" "6#/,&%\"yG\"\"\"\"\"#!\"\"%!G" }{TEXT 320 1 "+ " }{TEXT -1 1 " " }{XPPEDIT 18 0 "4*sqrt(2)/3" "6#*(\"\"%\"\"\"-%%sqrt G6#\"\"#F%\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 4 "T he " }{TEXT 322 1 "y" }{TEXT -1 39 " intercepts occur at the points wh ere " }{XPPEDIT 18 0 "y=2" "6#/%\"yG\"\"#" }{TEXT -1 1 " " }{TEXT 321 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "4*sqrt(2)/3;" "6#*(\"\"%\"\" \"-%%sqrtG6#\"\"#F%\"\"$!\"\"" }{TEXT -1 12 ", so that, " }{TEXT 324 1 "y" }{TEXT -1 1 " " }{TEXT 323 1 "~" }{TEXT -1 13 " 0.1144 and " } {TEXT 325 1 "y" }{TEXT -1 1 " " }{TEXT 326 1 "~" }{TEXT -1 9 " 3.8856. " }}{PARA 0 "" 0 "" {TEXT -1 65 "These values make sense in connectio n with the previous picture. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "(x-1)^2/9+(y-2)^2/4=1:\neval (%,y=0); solve(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&\"\"*!\"\", &%\"xG\"\"\"F*F'\"\"#F*F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$\"\" \"F#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "(x-1)^2/9+(y-2)^2/4=1:\neval(%,x=0); solve(%); evalf[ 5](evalf(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&#\"\"\"\"\"*F&*&\" \"%!\"\",&%\"yGF&\"\"#F*F-F&F&" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$,&\" \"#\"\"\"*(\"\"%F%\"\"$!\"\"F$#F%F$F%,&F$F%*(F'F%F(F)F$F*F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"&c)Q!\"%$\"%W6F%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 "" {TEXT 299 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 41 "Sketch th e ellipse given by the equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "(x+1)^2/9+(y+3)^2/25 = 1;" "6#/,&*&,&%\"xG\"\"\"F(F(\" \"#\"\"*!\"\"F(*&,&%\"yGF(\"\"$F(F)\"#DF+F(F(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 51 "Identify the foci of the ellipse and dete rmine any " }{TEXT 297 1 "x" }{TEXT -1 5 " and " }{TEXT 298 1 "y" } {TEXT -1 13 " intercepts. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 300 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 27 "The ellipse with equation " }{XPPEDIT 18 0 "x^2/9+y^2/25 ;" "6#,&*&%\"xG\"\"#\"\"*!\"\"\"\"\"*&%\"yGF&\"#DF(F)" }{TEXT -1 5 " h as " }{TEXT 304 1 "x" }{TEXT -1 12 " intercepts " }{XPPEDIT 18 0 "x=`` " "6#/%\"xG%!G" }{TEXT 302 1 "+" }{TEXT -1 7 " 3 and " }{TEXT 303 1 "y " }{TEXT -1 13 " intercepts " }{XPPEDIT 18 0 "y=``" "6#/%\"yG%!G" } {TEXT 301 1 "+" }{TEXT -1 4 " 5. " }}{PARA 0 "" 0 "" {TEXT -1 8 "Letti ng " }{XPPEDIT 18 0 "c = sqrt(25-9);" "6#/%\"cG-%%sqrtG6#,&\"#D\"\"\" \"\"*!\"\"" }{XPPEDIT 18 0 "`` = 4;" "6#/%!G\"\"%" }{TEXT -1 22 ", the foci are on the " }{TEXT 327 1 "y" }{TEXT -1 22 " axis at the points \+ at" }{XPPEDIT 18 0 "``(0,-c);" "6#-%!G6$\"\"!,$%\"cG!\"\"" }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(0,c);" "6#-%!G6$\"\"!%\"cG" }{TEXT -1 13 " , that is, at" }{XPPEDIT 18 0 "``(0,-2);" "6#-%!G6$\"\"!,$\"\"#!\"\"" }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(0, 2);" "6#-%!G6$\"\"!\"\"#" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 26 "The ellipse with equati on " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x+1)^2/9+(y+3 )^2/25 = 1;" "6#/,&*&,&%\"xG\"\"\"F(F(\"\"#\"\"*!\"\"F(*&,&%\"yGF(\"\" $F(F)\"#DF+F(F(" }{TEXT -1 12 " ------- (i)" }}{PARA 0 "" 0 "" {TEXT -1 82 "is obtained by translating the first ellipse so that the centre moves to the point" }{XPPEDIT 18 0 "``(-1,-3);" "6#-%!G6$,$\"\"\"!\" \",$\"\"$F(" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 476 476 476 {PLOTDATA 2 "67-%'CURVESG6%7S7$$\"\"$\"\"!$F*F*7 $$\"3UF$)=7(3>(H!#<$\"3b@$)y74REo!#=7$$\"3(Q$pkH19-HF/$\"3aiuzzWkm7F/7 $$\"3i5$eO%fduFF/$\"3!o;GC;b:!>F/7$$\"3%z0RcF@ff#F/$\"3Kdnsp/@1DF/7$$ \"3s;D')4cFrBF/$\"3Ow9,NGziIF/7$$\"3v/=f]AxC@F/$\"3i\"=Y/Nr(HNF/7$$\"3 w)z&*QsO\\$=F/$\"3A21wC&fc&RF/7$$\"3UGA*oYJK]\"F/$\"3t'f%QpI,FVF/7$$\" 3IZ7S$*=/Y6F/$\"3];+`bFy?YF/7$$\"3/,WmG)ehd(F2$\"3v#pSs1Mz$[F/7$$\"30q `%yY5a.%F2$\"3g)zr=oeX&\\F/7$$!3+6F@K4K\"z\"!#?$\"3ik!e'3\"*****\\F/7$ $!3A%\\Cx2Zu3%F2$\"3WM(4QktL&\\F/7$$!3hn92^kaQzF2$\"3&3'4fuiw@[F/7$$!3 j8$>$zE$>8\"F/$\"3)**Q6zEL/j%F/7$$!3B\\n3#e%p7:F/$\"3:@$Rl`UyJ%F/7$$!3 j=o!H%o%=\"=F/$\"3S\"3*>>]6&)RF/7$$!3_t#HS!\\zE@F/$\"3]zP=IgQENF/7$$!3 Wudmh7ErBF/$\"3Eae:#pBG1$F/7$$!3M%3V>-#z(f#F/$\"3CZmu/-#3]#F/7$$!3-O4% >)3GpFF/$\"3wLV(*fk(G#>F/7$$!31s6#G0(3**GF/$\"3uYtI8>#fG\"F/7$$!3'H4De +4@(HF/$\"3,:-ED]7-oF27$$!3Cy\\ysb)***HF/$\"3M7hC=Zl]:!#>7$$!3m:ph09`s HF/$!3_ja*4Gf1v'F27$$!3glgA!HF/$!3mnx89:9m7F/7$$!3kX*4Rmy*yFF/$!3! [Df\"oMl*47e\\F/7$$!3iQO5 )G]@<*Fbo$!3+.='4jw***\\F/7$$\"3%3p'*3wpJ'QF2$!3atBaA4Pe\\F/7$$\"3!oz6 r.C^m(F2$!3gTUt9)RS$[F/7$$\"3Altm:&R2:\"F/$!3W^`3'fQvh%F/7$$\"3g$GHG^!G$=F/$!39xW#fk.%eRF/7$$\"3B\"GSf6! zH@F/$!3\\&G%HI5O@NF/7$$\"31el6=N0qBF/$!3C`Y'eF>a1$F/7$$\"3)>/j[D,Bg#F /$!3%)4**G'[jx[#F/7$$\"3u\\Z'fG+vw#F/$!3\"yf`$)G%)*H>F/7$$\"3qfowgmS'* GF/$!3)*Hv$)>>f-8F/7$$\"3mt'*ygm3sHF/$!3;q[tIl$[!oF27$F($\"3uAmIq&o?5% !#E-%'COLOURG6&%$RGBG$\"*++++\"!\")F+F][l-%*THICKNESSG6#\"\"\"-F$6&7$7 $F+$!\"%F*7$F+$\"\"%F*Fiz-%'SYMBOLG6$%'CIRCLEG\"#5-%&STYLEG6#%&POINTG- F$6&Ff[lFiz-F^\\l6$%(DIAMONDGFa\\lFb\\l-F$6&Ff[lFiz-F^\\l6$%&CROSSGFa \\lFb\\l-F$6&Ff[l-Fjz6&F\\[lF*F*F*-F^\\l6$F`\\l\"#7Fb\\l-F$6%7S7$$\"\" #F*$!\"$F*7$$\"3UF$)=7(3>(>F/$!3'z;@(34OF/$!3ZPD?? bNLF/7$$!3o7A$4K\"z,5F/$\"3ik!e'3\"**** *>F/7$$!3`\\Cx2Zu39F/$\"3WM(4QktL&>F/7$$!3'o92^kaQz\"F/$\"3&3'4fuiw@=F /7$$!3j8$>$zE$>8#F/$\"3)**Q6zEL/j\"F/7$$!3B\\n3#e%p7DF/$\"3:@$Rl`UyJ\" F/7$$!3j=o!H%o%=\"GF/$\"3+93*>>]6&)*F27$$!3_t#HS!\\zEJF/$\"3'\\zP=IgQE &F27$$!3Wudmh7ErLF/$\"31Eae:#pBG'Fas7$$!3M%3V>-#z(f$F/$!3kFN`_zz\"*\\F 27$$!3-O4%>)3GpPF/$!3Cmc-SN7x5F/7$$!31s6#G0(3**QF/$!3F`Ep'3ySr\"F/7$$! 3'H4De+4@(RF/$!3GyRZ(\\(y>BF/7$$!3Cy\\ysb)***RF/$!33Rv\"GX$\\%)HF/7$$! 3m:ph09`sRF/$!3cY&*4Gf1vOF/7$$!3glgA!RF/$!35ox89:9mUF/7$$!3kX*4Rmy *yPF/$!3ca#f\"oMl*47ezF/ 7$$!3q.\")G]@<45F/$!3+.='4jw***zF/7$$!3;4L5R-$o8'F2$!3atBaA4PezF/7$$!3 ?.#))G'f([L#F2$!3gTUt9)RS$yF/7$$\"34_Onc^R2:F2$!3W^`3'fQvh(F/7$$\"3)f$ GHG^!G$)F2$!39xW#fk.%epF/7$$\"3B\"GS f6!zH6F/$!3/&G%HI5O@lF/7$$\"31el6=N0q8F/$!3C`Y'eF>a1'F/7$$\"3)>/j[D,Bg \"F/$!3%)4**G'[jx[&F/7$$\"3u\\Z'fG+vw\"F/$!3.)f`$)G%)*H\\F/7$$\"3qfowg mS'*=F/$!3?Iv$)>>f-VF/7$$\"3mt'*ygm3s>F/$!3+([tIl$[!o$F/7$F[^l$!3aJz*e *******HF/-Fjz6&F\\[l$Fa\\l!\"\"F+F+-Fa[l6#F\\^l-F$6&7$7$$F`]mF*$!\"(F *7$Fg]m$Fc[lF*-Fjz6&F\\[lF+F][lF+F]\\lFb\\l-F$6&Fe]mF\\^mFh\\lFb\\l-F$ 6&Fe]mF\\^mF]]lFb\\l-F$6&Fe]mFb]lFd]lFb\\l-F$6&7#7$Fg]mF]^l-F^\\l6#F` \\lFb]lFb\\l-F$6&Ff^m-F^\\l6#Fj\\lFb]lFb\\l-F$6&Ff^m-F^\\l6#F_]lFb]lFb \\l-F$6%7$7$Fh[lF]^lFj]lFb]l-%*LINESTYLEG6#F)-F$6%7$7$Fg]m$F_[lF*7$Fg] mF[^lFb]lFf_m-%%TEXTG6%7$$!#n!\"#$!$x#Fe`mQ&(1,2)6\"-%&COLORG6&F\\[l$F c[lFe`mF]amF]am-F``m6%7$$\"#NF`]m$Fe`mF`]mQ\"xFi`mFj`m-F``m6%7$Fcam$\" #bF`]mQ\"yFi`mFj`m-%+AXESLABELSG6%Q!Fi`mF^bm-%%FONTG6#%(DEFAULTG-%(SCA LINGG6#%,CONSTRAINEDG-%%VIEWG6$;$!#XF`]mFaam;$!#&)F`]mFham" 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" "Curve 17" "Curve 18" }}{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 38 "The foci of the translated ellipse are" }{XPPEDIT 18 0 "``(-1, \+ 1);" "6#-%!G6$,$\"\"\"!\"\"F'" }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(- 1, -7);" "6#-%!G6$,$\"\"\"!\"\",$\"\"(F(" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 12 "To find any " }{TEXT 305 1 "x" }{TEXT -1 42 " inte rcepts of the translated ellipse put " }{XPPEDIT 18 0 "y=0" "6#/%\"yG \"\"!" }{TEXT -1 18 " in equation (i). " }}{PARA 0 "" 0 "" {TEXT -1 12 "This gives: " }}{PARA 256 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 " (x+1)^2/9+9/25 = 1;" "6#/,&*&,&%\"xG\"\"\"F(F(\"\"#\"\"*!\"\"F(*&F*F( \"#DF+F(F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x+1)^2/9 = 16/25;" " 6#/*&,&%\"xG\"\"\"F'F'\"\"#\"\"*!\"\"*&\"#;F'\"#DF*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x+1)/3 = ``;" "6#/*&,&%\"xG\"\"\"F'F'F'\"\"$!\"\"% !G" }{TEXT 328 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "4/5" "6#*&\"\"%\" \"\"\"\"&!\"\"" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x+1=``" "6#/,&% \"xG\"\"\"F&F&%!G" }{TEXT 331 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "12 /5" "6#*&\"#7\"\"\"\"\"&!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 330 1 "x" } {TEXT -1 39 " intercepts occur at the points where " }{XPPEDIT 18 0 " x = -1;" "6#/%\"xG,$\"\"\"!\"\"" }{TEXT -1 1 " " }{TEXT 329 1 "+" } {TEXT -1 1 " " }{XPPEDIT 18 0 "12/5;" "6#*&\"#7\"\"\"\"\"&!\"\"" } {TEXT -1 11 ", so that, " }{XPPEDIT 18 0 "x = 7/5;" "6#/%\"xG*&\"\"(\" \"\"\"\"&!\"\"" }{XPPEDIT 18 0 "``=1.4" "6#/%!G-%&FloatG6$\"#9!\"\"" } {TEXT -1 4 " or " }{XPPEDIT 18 0 "x = -17/5;" "6#/%\"xG,$*&\"#<\"\"\" \"\"&!\"\"F*" }{XPPEDIT 18 0 "``=-3.4" "6#/%!G,$-%&FloatG6$\"#M!\"\"F* " }{TEXT -1 3 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "To find any " }{TEXT 306 1 "y" }{TEXT -1 42 " intercepts \+ of the translated ellipse put " }{XPPEDIT 18 0 "x = 0;" "6#/%\"xG\"\"! " }{TEXT -1 18 " in equation (i). " }}{PARA 0 "" 0 "" {TEXT -1 12 "Thi s gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "1/9+(y+3 )^2/25 = 1;" "6#/,&*&\"\"\"F&\"\"*!\"\"F&*&,&%\"yGF&\"\"$F&\"\"#\"#DF( F&F&" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(y+3)^2/25 = 8/9;" "6#/*&,& %\"yG\"\"\"\"\"$F'\"\"#\"#D!\"\"*&\"\")F'\"\"*F+" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "(y+3)/5 = ``;" "6#/*&,&%\"yG\"\"\"\"\"$F'F'\"\"&!\" \"%!G" }{TEXT 332 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "2*sqrt(2)/3;" "6#*(\"\"#\"\"\"-%%sqrtG6#F$F%\"\"$!\"\"" }{TEXT -1 2 ", " }}{PARA 0 " " 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y+3 = ``;" "6#/,&%\"yG\"\"\"\"\"$F&%!G" }{TEXT 335 1 "+ " }{TEXT -1 1 " " }{XPPEDIT 18 0 "10*sqrt(2)/3;" "6#*(\"#5\"\"\"-%%sqr tG6#\"\"#F%\"\"$!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 334 1 "x" }{TEXT -1 39 " \+ intercepts occur at the points where " }{XPPEDIT 18 0 "y = -3;" "6#/% \"yG,$\"\"$!\"\"" }{TEXT -1 1 " " }{TEXT 333 1 "+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "10*sqrt(2)/3;" "6#*(\"#5\"\"\"-%%sqrtG6#\"\"#F%\"\"$!\" \"" }{TEXT -1 11 ", so that, " }{TEXT 339 1 "x" }{TEXT -1 1 " " } {TEXT 338 1 "~" }{TEXT -1 11 " 1.7140 or " }{TEXT 337 1 "x" }{TEXT -1 1 " " }{TEXT 336 1 "~" }{TEXT -1 10 " -7.7140. " }}{PARA 0 "" 0 "" {TEXT -1 87 "Both the x and y intercepts appear reasonable in connecti on with the previous picture. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "(x+1)^2/9+(y+3)^2/25=1:\neva l(%,y=0); solve(%); evalf[5](evalf(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&\"\"*!\"\",&%\"xG\"\"\"F*F*\"\"#F*#F&\"#DF*F*" }}{PARA 11 " " 1 "" {XPPMATH 20 "6$#\"\"(\"\"&#!# " 0 "" {MPLTEXT 1 0 66 "(x+1)^2/9+(y+3)^2/25=1:\neval(%,x=0 ); solve(%); evalf[5](evalf(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/, &#\"\"\"\"\"*F&*&\"#D!\"\",&%\"yGF&\"\"$F&\"\"#F&F&" }}{PARA 11 "" 1 " " {XPPMATH 20 "6$,&\"\"$!\"\"*(\"#5\"\"\"F$F%\"\"##F(F)F(,&F$F%*(F'F(F $F%F)F*F%" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"&Sr\"!\"%$!&Sr(F%" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 70 "Translating a hyperbola so that its centr e moves away from the origin " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 476 476 476 {PLOTDATA 2 "6?-%'CURVESG6%7U7$$\"3+A%zhR&p_k!#< $!3>oI!H+.'3QF*7$$\"3T9OE*y@_.'F*$!3MXsWR!*=\"\\$F*7$$\"31_'*G0Bl+dF*$ !3@&pk5J5;B$F*7$$\"3?<,.%RWVN&F*$!37Dm/[+lcHF*7$$\"3b(fK&zl)e.&F*$!3iy 3j]C^'p#F*7$$\"3e:ZL'z#>ZZF*$!3gFmA3!RFX#F*7$$\"3'z6zt/,L]%F*$!3(y_@)e '>!RAF*7$$\"3A1A/,ldtUF*$!3oKGua<1H?F*7$$\"3;6v'fnj\"fSF*$!3k'G1b/![;Tui\"F*7$$\"37#QP;81Hp$F*$!3(RohBllcV\" F*7$$\"3C4-zsnpcNF*$!3)*e&z#=4ot7F*7$$\"31\"QVGCtCU$F*$!3??.[:X9)4\"F* 7$$\"35Srcm6L2LF*$!3/b&4Osc=G*!#=7$$\"3ndPR[fO9KF*$!3!yr[fO#e%p(Fho7$$ \"3w:x`vA#[9$F*$!3qF.9eE**)G'Fho7$$\"3!=BpX*R7!3$F*$!3cWa)pXTJl%Fho7$$ \"3BiPaT(H./$F*$!3=NBl#RB/H$Fho7$$\"3)=Og*)[;6,$F*$!3L8J:(oOLs\"Fho7$$ \"38]()*f!oW+IF*$!3E7y8FX\">X$!#>7$$\"3tCoYar30IF*$\"3?/`VZW@l6Fho7$$ \"3qp:n1$Ga-$F*$\"3w[2nlZ_4EFho7$$\"3E!Gk$*=EL1$F*$\"3\"H)Rail+JTFho7$ $\"3uiaRTgI8JF*$\"3COF_!Hr%[bFho7$$\"35\">4k+(y$=$F*$\"3$[$yh-g62rFho7 $$\"3YGxA8mhvKF*$\"31\"=%*f*)zxw)Fho7$$\"3SDVT_+UrLF*$\"3]\\$F*$\"3[N[e.:Y\">\"F*7$$\"3eRehGu)ej$F*$\"3I=H\\!*oXp8F *7$$\"3M]ZcD@U'z$F*$\"3e*4\"4N8.^:F*7$$\"3mGySI^/[@YkML0B!z[>F*7$$\"3z!*oPd+D1WF*$\"3?KI)>O*[^@F*7$ $\"3?0&oc&G'Hm%F*$\"3)=^VUPV)zBF*7$$\"3K$*[&e'3b=\\F*$\"3aw-'*zKZ)f#F* 7$$\"3lY2&e+DWA&F*$\"3%Rza_W![^GF*7$$\"3,DBLMx%*QbF*$\"3+%\\B_(\\6/JF* 7$$\"3C7bJ%RNr*eF*$\"3o]i%R%[o%Q$F*7$$\"3sOLdX/CyiF*$\"3)\\:7t#=twOF*7 $$\"3'y+8o.O@r'F*$\"3@X#GHgIH+%F*7$$\"3)>#QqT&fg;(F*$\"3W5&ol\">eQVF*7 $$\"3mb\"*)**f\\$pwF*$\"3!)4pPF9]0ZF*7$$\"330B2\"3@-@)F*$\"3=f^-$o'*\\ 4&F*7$$\"3K4KM<#)RY()F*$\"3o`(>'\\X?xaF*7$$\"3H@cFzcJ5%*F*$\"3yLTb4[?Y fF*7$$\"3Ht]G9^>05!#;$\"3=1-w)z\"*eR'F*7$$\"3Y%*Q-%=T*y5Fhy$\"3%QlidL) H4pF*7$$\"30u-#>,&H;6Fhy$\"3_@=r,X=orF*7$$\"3Mo:SNl/b6Fhy$\"3LUiuRe/Ou F*7$$\"3cP8TyQI)>\"Fhy$\"3@P62HyGMxF*7$$\"3&[4B6&RHV7Fhy$\"3[qYJ%[8P/) F*-%'COLOURG6&%$RGBG$\"*++++\"!\")$\"\"!F\\\\lFh[l-%*THICKNESSG6#\"\"# -F$6%7W7$$!3+A%zhR&p_kF*F+7$$!35d)z*[zL\"G'F*$!3_k`W*R#3zOF*7$$!3GZ#3' 3.$e6'F*$!33q'\\N7))Hb$F*7$$!3y8>0s@YwfF*$!3'Rz@uFqfW$F*7$$!30BEY%o28% eF*$!3v>#H0)>QTLF*7$$!3/m7umaA_bF*$!3gm.#R\")RY6$F*7$$!3;J'ywSI7G&F*$! 3UjRblah(*GF*7$$!3!>!=Vl4[I]F*$!3m\\:rIV-#p#F*7$$!3d6Fr9X69[F*$!3RcRBE >/5DF*7$$!3TMz$=g&o0YF*$!3]*yTF*QuHBF*7$$!3#R)e-ur51WF*$!3mmSc/$f8:#F* 7$$!3oB3S#HYEA%F*$!3gOwF*7$$!3#zj[gfy$\\SF*$!3&[iDK**=K\"=F*7$ $!3%pr3`,I#4RF*$!3%RqYRVQ4n\"F*7$$!3'e`0Lf,\\w$F*$!3340.z%*\\;:F*7$$!3 (*3vTw,!Qj$F*$!3]loJOC*pO\"F*7$$!3Ec&4g!>6?NF*$!3W!4Q\"[5\"F*7$$!3E\"4$eS9[HLF*$!3<0>mvPQF'*Fho7$$!3] 5judo+dKF*$!35Y&z*=A/a%)Fho7$$!3i%3yIg,W=$F*$!3cjC%f>V$>rFho7$$!3)e(Qj PzTIJF*$!3epSJOc/hfFho7$$!3%R#zA/v4#3$F*$!3Y!=hLFi3r%Fho7$$!39Lamiy`YI F*$!3$*ow,-$F*$!3U)*3[<$[MK#Fho7$$!3HMU(G<\\b+ $F*$!3Z%Q4+DCq@\"Fho7$$!3f9s4lG++IF*$!3MF@gYD5kF!#?7$$!3S(y5#y\\Y0IF*$ \"3iS$)yiyu27Fho7$$!3RD0w_da>IF*$\"31Rk)p1SnG#Fho7$$!3X`<()e@bWIF*$\"3 !4L#HwVffMFho7$$!3meH;45;\"3$F*$\"3U(Rd[$fb$o%Fho7$$!3+=6GmspFJF*$\"3% H3GFqJs*eFho7$$!3'**o/MYoH=$F*$\"3q*e!H-B!34(Fho7$$!3+\"pK2+ijD$F*$\"3 \"\\ZJIT%*HW)Fho7$$!3HSFGbQLLLF*$\"3?T\"['z+X'o*Fho7$$!3$ecP/UDtU$F*$ \"3o\"*RQ@)\\[5\"F*7$$!3dhAqr!3L_$F*$\"35W%z`oY5Nsu8F*7$$!3'3G#*[&)4Cw$F*$\"3*))yFd`[P^\"F*7$$!3bW()4oRx-RF*$ \"3yfwf'[F*7$$!3?'\\%*GKm[S%F*$\"3c'fa^oH-:#F*7$$!3$G`A$)y(G/YF*$ \"3Ur6L,c^GBF*7$$!3St4))es4=[F*$\"3x(G*3,gV8DF*7$$!36BR&H+&GH]F*$\"3mN s_**4.\"p#F*7$$!3Atm)*f\")[*G&F*$\"33h3Us@I/HF*7$$!3==*fuYI%RbF*$\"3`! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 51 "Similarly, translating the hyperbola \+ with equation " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^2 /(a^2)-x^2/(b^2) = 1;" "6#/,&*&%\"yG\"\"#*$%\"aGF'!\"\"\"\"\"*&%\"xGF' *$%\"bGF'F*F*F+" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 53 "so th at its centre moves from the origin to the point" }{XPPEDIT 18 0 "``(h ,k)" "6#-%!G6$%\"hG%\"kG" }{TEXT -1 36 " gives the hyperbola with equa tion: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(y-k)^2/(a^ 2)-(x-h)^2/(b^2) = 1;" "6#/,&*&,&%\"yG\"\"\"%\"kG!\"\"\"\"#*$%\"aGF+F* F(*&,&%\"xGF(%\"hGF*F+*$%\"bGF+F*F*F(" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{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 84 "Q1. Sketch each of the fo llowing ellipses and identify the foci. Also determine any " }{TEXT 262 1 "x" }{TEXT -1 5 " and " }{TEXT 263 1 "y" }{TEXT -1 13 " intercep ts. 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " (a) " }{XPPEDIT 18 0 "(x+1)^2/4-(y-2)^2/9 = 1;" "6#/,&*& ,&%\"xG\"\"\"F(F(\"\"#\"\"%!\"\"F(*&,&%\"yGF(F)F+F)\"\"*F+F+F(" } {TEXT -1 14 " (b) " }{XPPEDIT 18 0 "(x+3)^2/16-(y+2)^2/25 = 1 " "6#/,&*&,&%\"xG\"\"\"\"\"$F(\"\"#\"#;!\"\"F(*&,&%\"yGF(F*F(F*\"#DF,F ,F(" }{TEXT -1 13 " (c) " }{XPPEDIT 18 0 "(y-1)^2/4-(x+2)^2 = \+ 1" "6#/,&*&,&%\"yG\"\"\"F(!\"\"\"\"#\"\"%F)F(*$,&%\"xGF(F*F(F*F)F(" } {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " (d) " }{XPPEDIT 18 0 "(y-2)^2/4-(x-1)^2/9 = 1" "6#/,&*&, &%\"yG\"\"\"\"\"#!\"\"F)\"\"%F*F(*&,&%\"xGF(F(F*F)\"\"*F*F*F(" }{TEXT -1 14 " (e) " }{XPPEDIT 18 0 "(x+2)^2/16-(y-3)^2/9 = 1" "6#/, &*&,&%\"xG\"\"\"\"\"#F(F)\"#;!\"\"F(*&,&%\"yGF(\"\"$F+F)\"\"*F+F+F(" } {TEXT -1 13 " (f) " }{XPPEDIT 18 0 "(x+1)^2/16-(y+2)^2/25 = 1 " "6#/,&*&,&%\"xG\"\"\"F(F(\"\"#\"#;!\"\"F(*&,&%\"yGF(F)F(F)\"#DF+F+F( " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " (g) " }{XPPEDIT 18 0 "(y-3)^2-(x-3)^2=1" "6#/,&*$,&%\"yG \"\"\"\"\"$!\"\"\"\"#F(*$,&%\"xGF(F)F*F+F*F(" }{TEXT -1 16 " \+ (h) " }{XPPEDIT 18 0 "(y+2)^2-(x+2)^2=1" "6#/,&*$,&%\"yG\"\"\"\"\"#F (F)F(*$,&%\"xGF(F)F(F)!\"\"F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 9 "Ans (a) " }} {PARA 0 "" 0 "" {TEXT -1 6 " (a) " }{XPPEDIT 18 0 "(x+1)^2/4-(y-2)^2/ 9 = 1;" "6#/,&*&,&%\"xG\"\"\"F(F(\"\"#\"\"%!\"\"F(*&,&%\"yGF(F)F+F)\" \"*F+F+F(" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "p1 := plots[implicitplot]((x+1)^2/ 4-(y-2)^2/9=1,x=-9..7,y=-6..10,scaling=constrained):\np2 := plot([3*(x +1)/2+2,-3*(x+1)/2+2],x=-9..7,color=black,linestyle=3):\nplots[display ](p1,p2);\n\n" }}{PARA 13 "" 1 "" {GLPLOT2D 466 466 466 {PLOTDATA 2 "6 (-%'CURVESG6ir7$7$$!3#*oC!e8puo'!#<$!\"'\"\"!7$$!3#[A**of(Q&f'F*$!3cv2 5.ChWeF*7$7$$!3]************RkF*$!3)QLLL$eR#e&F*F.7$7$$!3R++++++SkF*$ \"3yMLLLeR#e*F*7$F($\"#5F-7$7$F;F77$$!39nmmm;/^jF*$!3eLLLL$e*[aF*7$7$$ !3k6666ht\"H'F*$!3K++++++g`F*FD7$FJ7$$!3zz8C()F*7$Feo7$$!33)********\\(egF*$\"3C'******* *\\(y*)F*7$7$$!3u5666ht\"H'F*$\"3U************f$*F*F[p7$7$$!3')4666ht \"H'F*FdpF:7$F\\o7$$!3Mb[];hz@cF*$!3AZ^\\$)Q?eUF*7$7$$!3C%z]Oz]'3bF*$! 3'4++++++3%F*F[q7$Faq7$$!3-)ooooo=Q&F*$!3'[JJJJJ\"eQF*7$7$$!3/,+++++g^ F*$!3i76666')pMF*Fgq7$7$F^r$\"3186666')puF*7$$!3Z#z]Oz]'3bF*$\"3I)**** *******z!)F*7$7$$!3N$z]Oz]'3bF*FirFbo7$7$F^r$!3186666')pMF*7$$!3]cbbbb 0[^F*$!3#eWWWWW>X$F*7$7$$!3)Gf#f#f#4S^F*$!3$3++++++W$F*Fcs7$Fis7$$!35! 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{XPPMATH 20 "6$,&\"\"#\"\"\"*(F$F%\"\"$!\"\"\"#5#F%F$F%,&F$F%*(F$F%F 'F(F)F*F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"T!\"%$!%#3\"F%" }}} {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 "" }}{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 0 "" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 18 "Code for pic tures " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 32 "Circle with centre at the origin" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 665 "sn := 0.5: cs := evalf(cos(Pi/6)):\np1 := plot([cos(t),sin(t),t=0 ..2*Pi],thickness=2):\np2 := plot([[0,0],[cs,sn]],color=COLOR(RGB,0,.8 ,0),thickness=2):\np3 := plot([[[cs,sn]]$3],style=point,symbol=[circle ,diamond,cross],color=black):\np4 := plot([[[0,0]]$4],style=point,symb ol=[circle,diamond,cross,circle],\n color=[green$3,black],symbolsize =[10$3,12]):\nt1 := plots[textplot]([[1.15,-.05,`x`],[-0.05,1.15,`y`], [1.07,.53,`P(x,y)`],\n [-.13,.11,`(0,0)`]],color=black,font=[HELVETIC A,10]):\nt2 := plots[textplot]([[.42,.35,`a`]],color=black,font=[TIMES ,ITALIC,11]):\nplots[display]([p||(1..4),t1,t2],tickmarks=[0,0],view=[ -1.15..1.15,-1.15..1.15],\n scaling=constrained);" }}}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 0 "" 0 "" {TEXT -1 34 "Ellipse which is close to a circl e" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 893 "aa := 1: bb := evalf(sqrt(99)/10): cc := 1/10:\np1 : = plot([aa*cos(t),bb*sin(t),t=0..2*Pi],thickness=2,color=red):\np2 := \+ plot([[[-cc,0],[cc,0]]$4],style=point,symbol=[circle,diamond,cross,cir cle],\n color=[green$3,black],symbolsize=[10$3,12]):\ntt := 1.3:\np3 := plot([[-cc,0],[aa*cos(tt),bb*sin(tt)],[cc,0]],color=COLOR(RGB,0,.8 ,0)):\np4 := plot([[[aa*cos(tt),bb*sin(tt)]]$3],style=point,symbol=[ci rcle,diamond,cross],\n color=black,symbolsize=[10$3]):\nt1 := plots [textplot]([[-.18,.14,`F`],[.33,1.08,`P(x,y)`],[.22,.14,`F`]],\n fon t=[HELVETICA,10],color=COLOR(RGB,.01,.01,.01)):\nt2 := plots[textplot] ([[-.14,.1,`1`],[.26,.1,`2`]],font=[HELVETICA,8],\n color=COLOR(RGB, .01,.01,.01)):\nt3 := plots[textplot]([[1.15,-.06,`x`],[-.06,1.15,`y`] ],font=[HELVETICA,9],\n color=COLOR(RGB,.01,.01,.01)):\nplots[displa y]([p||(1..4),t1,t2,t3],font=[HELVETICA,9],\n scaling=constrained,t ickmarks=[4,4]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 38 "Ci rcle with centre at the point (h,k) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 676 "sn := 0.5: cs := evalf(co s(Pi/6)):\np1 := plot([3+cos(t),2+sin(t),t=0..2*Pi],thickness=2):\np2 \+ := plot([[3,2],[3+cs,2+sn]],color=COLOR(RGB,0,.8,0),thickness=2):\np3 \+ := plot([[[3+cs,2+sn]]$3],style=point,symbol=[circle,diamond,cross],co lor=black):\np4 := plot([[[3,2]]$4],style=point,symbol=[circle,diamond ,cross,circle],\n color=[green$3,black],symbolsize=[10$3,12]):\nt1 : = plots[textplot]([[4.18,-.08,`x`],[-0.08,3.15,`y`],[4.17,2.56,`P(x,y) `],\n [2.87,2.2,`(h,k)`]],color=black,font=[HELVETICA,10]):\nt2 := pl ots[textplot]([[3.42,2.42,`a`]],color=black,font=[TIMES,ITALIC,11]):\n plots[display]([p||(1..4),t1,t2],tickmarks=[0,0],view=[-.1..4.18,-.1.. 3.15],\n scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1036 "sn := 0.5: cs := evalf(cos (Pi/6)):\np1 := plot([cos(t),sin(t),t=0..2*Pi],thickness=2):\np2 := pl ot([[0,0],[cs,sn]],color=COLOR(RGB,0,.8,0),thickness=2):\np3 := plot([ [[cs,sn]]$3],style=point,symbol=[circle,diamond,cross],color=black):\n p4 := plot([[[0,0]]$4],style=point,symbol=[circle,diamond,cross,circle ],\n color=[green$3,black],symbolsize=[10$3,12]):\np5 := plot([3+cos (t),2+sin(t),t=0..2*Pi],thickness=2):\np6 := plot([[3,2],[3+cs,2+sn]], color=COLOR(RGB,0,.8,0),thickness=2):\np7 := plot([[[3+cs,2+sn]]$3],st yle=point,symbol=[circle,diamond,cross],color=black):\np8 := plot([[[3 ,2]]$4],style=point,symbol=[circle,diamond,cross,circle],\n color=[g reen$3,black],symbolsize=[10$3,12]):\nt1 := plots[textplot]([[4.18,-.0 8,`x`],[-0.08,3.15,`y`],[1.24,.56,`(x-h,y-k)`],\n [2.87,2.2,`(h,k)`], [4.12,2.56,`(x,y)`]],\n color=black,font=[HELVETICA,10]):\nt2 := plo ts[textplot]([[.42,.42,`a`],[3.42,2.42,`a`]],color=black,font=[TIMES,I TALIC,11]):\nplots[display]([p||(1..8),t1,t2],tickmarks=[0,0],view=[-1 .1..4.18,-1.1..3.15],\n scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 24 "Translating an ellipse " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1149 "aa := 3: bb := evalf(sqrt(5)): cc := 2:\np1 := plot([aa*cos(t),bb*sin(t),t= 0..2*Pi],thickness=2):\np2 := plot([[[-cc,0],[cc,0]]$4],style=point,sy mbol=[circle,diamond,cross,circle],\n color=[green$3,black],symbolsi ze=[10$3,12]):\ntt := 1.3:\np3 := plot([[-cc,0],[aa*cos(tt),bb*sin(tt) ],[cc,0]],color=COLOR(RGB,0,.8,0)):\np4 := plot([[[aa*cos(tt),bb*sin(t t)]]$3],style=point,symbol=[circle,diamond,cross],\n color=black,sy mbolsize=[10$3]):\np5 := plot([6+aa*cos(t),4+bb*sin(t),t=0..2*Pi],thic kness=2):\np6 := plot([[[6-cc,4],[6+cc,4]]$4],style=point,symbol=[circ le,diamond,cross,circle],\n color=[green$3,black],symbolsize=[10$3,1 2]):\np7 := plot([[6-cc,4],[6+aa*cos(tt),4+bb*sin(tt)],[6+cc,4]],color =COLOR(RGB,0,.8,0)):\np8 := plot([[[6+aa*cos(tt),4+bb*sin(tt)]]$3],sty le=point,symbol=[circle,diamond,cross],\n color=black,symbolsize=[1 0$3]):\np9 := plot([[[3,4],[9,4]],[[6,4-bb],[6,4+bb]]],color=black,lin estyle=3):\nt1 := plots[textplot]([[1,2.6,`(x-h,y-k)`],[7,6.55,`(x,y)` ],[9.2,-.2,`x`],[-.15,6.73,`y`]],\n font=[HELVETICA,10],color=COLOR( RGB,.01,.01,.01)):\nplots[display]([p||(1..9),t1],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 30 "Translated ellipse examples " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 793 "h := 1: k := 2: a := 3: b := 2: c := evalf(sqrt(a^2-b^2)):\np1 := plot( [a*cos(t),b*sin(t),t=0..2*Pi],thickness=1,color=magenta):\np2 := plot( [[[-c,0],[c,0]]$4],style=point,symbol=[circle,diamond,cross,circle],\n color=[magenta$3,black],symbolsize=[10$3,12]):\np3 := plot([h+a*cos (t),k+b*sin(t),t=0..2*Pi],thickness=2):\np4 := plot([[[h-c,k],[h+c,k]] $4],style=point,symbol=[circle,diamond,cross,circle],\n color=[green $3,black],symbolsize=[10$3,12]):\np5 := plot([[[h,k]]$3],style=point,s ymbol=[circle,diamond,cross],color=black):\np6 := plot([[[h-a,k],[h+a, k]],[[h,k-b],[h,k+b]]],color=black,linestyle=3):\nt1 := plots[textplot ]([[h+.33,k+.23,`(1,2)`],[4.5,-.2,`x`],[-.2,4.5,`y`]],\n color=C OLOR(RGB,.01,.01,.01)):\nplots[display]([p||(1..6),t1],view=[-3.5..4.5 ,-2.5..4.5],scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 795 "h := -1: k := -3: a := 3: b := 5: c := evalf(sqrt(b^2-a^2)):\np1 := plot([a*cos(t),b*sin(t),t=0.. 2*Pi],thickness=1,color=magenta):\np2 := plot([[[0,-c],[0,c]]$4],style =point,symbol=[circle,diamond,cross,circle],\n color=[magenta$3,blac k],symbolsize=[10$3,12]):\np3 := plot([h+a*cos(t),k+b*sin(t),t=0..2*Pi ],thickness=2):\np4 := plot([[[h,k-c],[h,k+c]]$4],style=point,symbol=[ circle,diamond,cross,circle],\n color=[green$3,black],symbolsize=[10 $3,12]):\np5 := plot([[[h,k]]$3],style=point,symbol=[circle,diamond,cr oss],color=black):\np6 := plot([[[h-a,k],[h+a,k]],[[h,k-b],[h,k+b]]],c olor=black,linestyle=3):\nt1 := plots[textplot]([[h+.33,k+.23,`(1,2)`] ,[3.5,-.2,`x`],[-.2,5.5,`y`]],\n color=COLOR(RGB,.01,.01,.01)): \nplots[display]([p||(1..6),t1],view=[-4.5..3.5,-8.5..5.5],scaling=con strained);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 24 "Translat ing a hyperbola " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1259 "aa := 3: bb := 2: cc := sqrt(13):\np1 := plo t([aa*cosh(t),bb*sinh(t),t=-1.4..2.1],thickness=2,color=magenta):\np2 \+ := plot([-aa*cosh(t),bb*sinh(t),t=-1.4..1.4],thickness=2,color=magenta ):\np3 := plot([[[-cc,0],[cc,0]]$4],style=point,symbol=[circle,diamond ,cross,circle],\n color=[green$3,black],symbolsize=[10$3,12]):\ntt : = 1.3:\np4 := plot([[-cc,0],[aa*cosh(tt),bb*sinh(tt)],[cc,0]],color=CO LOR(RGB,0,.8,0)):\np5 := plot([[[aa*cosh(tt),bb*sinh(tt)]]$3],style=po int,symbol=[circle,diamond,cross],\n color=black,symbolsize=[10$3]) :\np6 := plot([6+aa*cosh(t),8+bb*sinh(t),t=-1.6..1.6],thickness=2):\np 7 := plot([6-aa*cosh(t),8+bb*sinh(t),t=-2.05..1.6],thickness=2):\np8 : = plot([[[6-cc,8],[6+cc,8]]$4],style=point,symbol=[circle,diamond,cros s,circle],\n color=[green$3,black],symbolsize=[10$3,12]):\ntt := 1.3 :\np9 := plot([[6-cc,8],[6+aa*cosh(tt),8+bb*sinh(tt)],[6+cc,8]],color= COLOR(RGB,0,.8,0)):\np10 := plot([[[6+aa*cosh(tt),8+bb*sinh(tt)]]$3],s tyle=point,symbol=[circle,diamond,cross],\n color=black,symbolsize= [10$3]):\nt1 := plots[textplot]([[4.9,4.1,`(x-h,y-k)`],[11.3,12.1,`(x, y)`],\n [13.6,-.3,`x`],[-.3,12.7,`y`]],font=[HELVETICA,10],color=COLO R(RGB,.01,.01,.01)):\nplots[display]([p||(1..10),t1],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 }