0002000201000aLadder.ACT000102000aLadder.EAC010000009ad8É Ladder-new.EAC† Œ.ACT† †#†-ˆ0†4† ІG ~I Ž'Á[ˆ4A lŠc problem†ŠX\Œ Author툈XíŽxŒ.Professor Wei-Chi Yang ŒLRadford University Ž9 e-mail: wy†9@rŒ2.eduŽE!URL: http://www.–#/Š5‡/ŽÖŽ Object†vs:[ެP1.†¥( will use ClassPad to explore a Calculus‘ that can be found in many 1@ctextbooks. And see how we can generate many different scenarios.[Q2. We will use its dynamic†Fometry a†bCAS features to help†2†b ke conjecŒŒX%before solving problems†L alytically.[Œ…Ž[†—ˆ˜ Example 1.Œ¨N A†nce q†‡et ta†² runs parallelˆ–a buildˆrat a†ûstaˆ6of pŒ9from ŽþWthe’2 . What is ˆlengthˆ8ˆ$shortest ladder tˆ*‰jreachŒTˆLgrou‡\+over the fence toŠ wall ofŠbuilding?[Note. ˆ #1For demonstration purpose, we useŒ^ollow†Fs:[Œ9$#1. We set p=4 and q=6 respectively.Ž+J2. T†E point E inœT Geometry Strip is¼coor†¼ ate (0,0).\Œ¶Example 1=>play‹animŠÂΈç†ëΆñŠ ††‰ 3X "‡ `63H‚`† ` cv"0#† b$Bf8Š ŽD‰>v’K†r‰PŽ A† 'LŽ'‰rC˜ u˜x††yŽ †Ž ÈÈ Ž'†-4fˆ8Q)u`Ž CAbPŽH ('€Ž$SWE€ Y5‡pŽeWv0r1H0yaY“@Ž`‡Vss0Žl–%SWŽ`xrtŽ)#gŽ„'’v‘Žœ@ˆ`ŽU@e•ލqyeŽ´A””ލyŠç6UpŠ×†¨eYt"Žä&CŽ$A„†T“)“ލX(‡pŽCU!Ž$X˜1  #$ƒ‘Da3‹ˆccccP‰”‹pDƒ‹o9@B‹3™f’(a0pŽ #$€`Ž‘&f Ž$`xf3y8@Ž<yaŽ0–’@Ž0‘y48Œ`†rQ0PŒls0Œxv‚ rh’ g3i„Žb€”•HXF˜d€Œ´T6I1ŽœPUu)`G62ŽØD!5SލA—I@e–ŽÀ@gbEBeb5Gi‡` W'!Àw(—Iü‰7D†‡OÄH ‹3‰\ˆÃH!‹EB‰4Ð6–°‡x%Šf – ІU†UŽCEC††Š” ž’ †3†8ÈÈ" †J)gQP†T Y™8Cg0Œ ca„ŠgY™„C$wŠ7Y™r`ŽGŽ7aHF€ŠH˜”’TuŠu ˜H˜Š ™ ™bŠ< ™IBXpŠ$™‰e`Š¥ ™B$2TŠ ™ ‰aQŽ SUЍ))h7 QX0@Œæv8T ŒòŠë5G8Ž$qˆˆŠÁ‡tŽ<Y ŠÀ'%d:y†i5‹AX™ ?V1RŽHx’$xŽœ 8)AŽx2ˆw‹«‘p`QuaX8w €Ž V) ‘@ŽT# U0Ž$R!P#y5$H1PP`ŽHFD“<DdyT`B‘”†lA'8Ql9r'™l8(xŽ6–)Yœ5y `4x„I´3˜SBÀ3Aƒ<30—´3t5À3eI’Ø4cPŽü6$(ð8eR@ðB2S’Ø‘!E0SUp‰<bux&‘Due™xT” pP8† ‡gÈÈ € ‹p†‡y(†'EŠòY™WC &Œ `e"‹RY™—‹ŽY™0rpŒ0#1CXY™%2PŒ ƒ…#`ŒY4E0Œ$U%1Fˆ1 ˜G`2tŠ ™‘4Ž F”'QpŒG58Œ0™b!sŽ BŠT#•8( IB$!Š`w—6gŒ$ 2–ЍF‡Ѝˆ–€ŒH7w8Ž•u'ŽHc"wRŠÌEq7%Ž<GU9IŽTvFV$ Œ D‰vg„Cr‹†† Y`T‹(WPG@Œ UC€vŽ,S@‘„Œ¼QB"t0IH"‹GYEXŽ\EvX#‹dD5#`B1dV‹ @qITŽx9!%€7—x@Ž 6Uyh`Ž5DgPŽ$4Q‰ Ž03x130q’€T3'U9ˆ_ˆ`3‡ 5wH6yBv$9C&E0ŽC$ŽœH34vœUEƒ‡e1•‚$yH1œ™VŽx† †âÈÈ €Š»ˆï†ô(†'EˆþY™WC &ŠnY™`e"ŠþY™—‹Y™0rpŒ0#1CX‰9†<%2‹"Y™ƒ…#Ž<Y4ŒÎY™U%1FŠl ˜G`2tŠ ™‘4Ž F”'QŠ` ™G58Šœ ™b!sŽ B"#•8(0IB$!Ž w—6gŒ 2–ˆ#†$F‡PŒ0ˆ–€Œ<7w8Ž•u'ŽHc"wR`Œ`Eq7%Ž<GU9IŽTvFV$ Œ„ D‰vg„CrŒ<†† Y`TŒ€WPG@Ž UC€vŒhS@‘„Œ¼QB"t0IH"pŽ<GYEX0EvX#D5#`B1dVŒø@qITx9!%Œø7—x„6Uyh„5Dg’¨4Q‰Œà3x1(30q’€T3'U9x3‡ˆýŒTw(6yBv9C&E0Ž C$ŽH34v@Ž$UEƒ‡PŽ0e1•‚$yH1 ŽH™VŽT† †_ÈÈ €Šˆl†q(†'Eˆ{Y™WC &ŠnY™`e"`Œ—ŠzY™0rpŒ0#1CXжY™%2Š’Y™ƒ…#Ž<Y4ŒÎY™U%1FŠl ˜G`2tŠ ™‘4Ž F”'QŠ` ™G58Šœ ™b!sŽ B"#•8( IB$!ŠÌw—6g: 2–^F‡:ˆ–€ŒH7w8R•u'ŽHc"wR‹Eq7%vGU9I²vFV$  D‰vgŒ Cr`І† Y`TPŠ WPG@Ž UC€vŽ,S@‘„Ž$QB"t0IH"pŽ<GYEXŽ\EvX#0ŽTD5#`B1dVˆ—ˆl@qITx9!%€Ž„7—x„6Uyhް5Dg’¨4Q‰Œà3x1À30q’€T3'U9x3‡ˆýŒTwH6yBvœ9C&EÀC$üH34v“ Eƒ‡Øe1•‚‘ yH1œ™VŽð† ‡’ÈÈ €‹›‰“‹‚(†'E‹¢Y™WC &Y™`e"`Ž —@Œ0rpŒ$#1CXˆ/†0%2PŒ<ƒ…#Ž<Y4E0ŒTU%1Fˆ` ˜G`2tŠ ™‘4Ž F”'QŠ` ™G58Œ0™b!sŽ BŠT#•8( IB$!Š`w—6gŠÌ 2–ˆ×†0F‡Ѝˆ–€ŒH7w8Ž•u'ŽHc"wR‹Eq7%Ž<GU9IŽTvFV$ Œ D‰vg„Cr‹D†† Y`T‹(WPG‹XUC€vŽ,S@‘„‹ˆQB"t0IH"‹|GYEXŽ\EvX#0D5#PŽ B1dVˆˆ@qIT9!%€Ž07—x@Ž<6Uyh`ŽH5DgH4Q‰ Ž`3x1`30q’€T3'U9x3‡„5wH6yBvœ9C&EÀC$ŽÌH34vœUEƒ‡Øe1•‚$yH1œ™VŽð† ‡ÈÈ €Š»‰‡$(†'E‰.Y™WC &ŠnY™`e"ŠþY™—‹Y™0rpŒ0#1CX‰i†<%2‹jY™ƒ…#Ž<Y4ŒÎY™U%1FŠl ˜G`2tŠ ™‘4Ž F”'Qp ™G58Œ b!sŽ B0ˆ$#•8( IB$!Š0w—6gŒ$ 2–ˆS†0F‡PŒ<ˆ–€ŒH7w8Ž•u'ŽHc"wR`ŒlEq7%Ž<GU9IŽTvFV$ Œ D‰vg„CrŒ<†† Y`TŒ€WPG@Œ UC€vŒhS@‘„Œ¼QB"t0IH"‹GYEX0EvX#D5#`B1dVŒø@qITx9!%Œø7—x„6Uyh„5Dg’¨4Q‰Œà3x1(30qŒø3€€3'U9ˆ Š ‡ 5w Ž$6yBv$9C&E0Ž<C$ŽHH34v@ŽTUEƒ‡PŽ`e1•‚$yH1T™VŽx††ŽÈÈ €Šsˆ›† (†'EˆªY™WC &ŠnY™`e"`Œ—ŠzY™0rpŒ0#1CXˆå†<%2Š’Y™ƒ…#Ž<Y4ŒÎY™U%1FŠl ˜G`2tŠ ™‘4Ž F”'QŠ` ™G58Šœ ™b!sŽ B"#•8( IB$!ŠÌw—6g: 2–‚F‡:ˆ–¦7w8R•u'c"wR`Œ Eq7%PŒGU9Iˆ#†$vFV$ Œ0 D‰vgŽ<CrŒ<†† Y`TŒDWPG@Ž UC€vŒhS@‘„Ž$QB"t0IH"pŽ<GYEX0EvX#0ŽTD5#`B1dVŒ¤@qITx9!%€Ž„7—x„6Uyh„5Dg’¨4Q‰Œà3x1À30q’€T3'U9x3‡ˆýŒTwH6yBvœ9C&EÀC$üH34v“ Eƒ‡Øe1•‚‘ yH1| ™V†† ÈÈ € І†(†'Eˆ'Y™WC &Œ `e"`Œ—@Œ$0rpŒ0#1CXŠbY™%2PŒHƒ…#Ž<Y4E0Œ`U%1FŠl ˜G`2tŠ ™‘4Ž F”'QŠ` ™G58Šœ ™b!sŽ BŠT#•8( IB$!ŠÌw—6gŠÌ 2– F‡Ѝˆ–€ŒH7w8Ž•u'ŽHc"wR‹Eq7%Ž<GU9I^vFV$‹W D‰vg„Cr‹D‹‚‹pY`T‹(WPG‹XUC€vŽ,S@‘„QB"tPŽ IH"pŽGYEX`Ž$EvX#0Ž0D5#0B1dVˆGˆH@qITH9!%€Ž`7—x@Žl6UyhT5Dgx4Q‰ Ž3x130q’€T3'U9x3‡„5wH6yBvœ9C&EÀC$üH34vœUEƒ‡Øe1•‚‘ yH1œ™VŽð†‡BÈÈ €Š»‰O‡T(†'EˆY™`e"‹FY™0r‹^Y™%2‹vY™Y4ŒžY™G`2tŠ  ™F”'QŠ0 ™b!sŽ #•8(0w—6gŒ F‡PŒ7w8Ž c"wR`Œ0GU9Iˆ;†< D‰vgŽHPQUƒŒ •3p2ŒTT`gsŒ<’'ƒŠ`pxŠ0`ƒ 4 Bp5Gu Œ0'@!Š„` 5‚vŠÀ`VTxˆË`•A7”ŠÌ`FƒT€Œ$0H™Ž tP˜Ž$††QY`TŒøUC€vŒìQB"tGYEXD5#0@qIT<7—x@Žø5Dg’T3x1L3€Š˜ˆ‡L6yBv C$‘@UEƒ‡”yH1 )A43PŒ h‚5@ŒaF„ €Š$'ƒpŽ0•‘‡ˆ;†<$U’pŒHWEBŒ01CRŒ0y")…0Š`eR13ŒTsugŽ•'hŒT%"†r`ŽH1W—ˆ¨ ™ˆˆŽ†¾9ÈÀ ŠÇ2ˆ¶Ð6–°†tR˜ˆ°E3•YˆŒ žC Length: ÿÿü†žc†’ŒÄÈ ‹2G˜k‰Eˆ `e(sV3‹VžK‡lÀHCH˜®“)“(˜¨¢žôõ 1@†Ð6–°†‡E!9ˆ ™9U˜ˆ ˆ*ˆ.Ž Coord: ÿÿü† †Ž†iÅH Šrˆ†{ÁH Œ„D˜„…‡CFQ4†Yˆ’Tžƒž†ÎÄ@ ’eˆ†ä ÈÈŽa–í”@†”â¤iŽKˆæÄH"”a†èŠaˆ{ˆ‰ˆ¢‹Œð!ŽðF™t‰‡†`–”¢ðœÊb‡· Ð6–°††† U‘ƒ† ‡–†† ††4Æ@  ˆ†Žˆ"ˆ†^ÅH ’*††t ËÈ#Œp@†„‹’r†˜—˜Ž(ˆKˆ‚Ę‹ˆˆaÿÿÿÿŽ"ˆq!†‹5†‹œµ"†ÿ ÈÀ˜Š™ ywR#©ˆÅH"“‹ A–ÍŒ+† `I33334†%”Z”³$¢ÉˆnˆŸ%ŠãH!œã"Ð $y1 ™†† † Y™ˆŒ ™y1D‚vˆ$˜†&†-Ç@  0†'†C ËÀ  Š@†RÐ6–°† ”W†’ ˜†€Š((†‰ÄH Š\C–GeF„18† ”F E)†CÄH#”ŸˆYˆŸ*Š˜¹ˆoˆ†ÿ+¢ˆ×,‡ÈȘ×rE%A†s;5$’6y‹G4•Bˆu`‡W-¢vŽX.‡vÆ@"’Ž/¢ªˆ40†Œ œŒÐ6–°††Š’ –`†'† †1†4Ä@  ˆ0†/dŽ"ˆ2Œ*!š*"†3†xÅ D4ˆŽÈÈ Œ@†Þ¤5†r!šœˆn4W–2g”‰f€†6¢ˆÙˆ7¢A'†8†™ †™Œóq˜š™2 ywR#‡U ‡˜²Ža†9‹H!œ{ˆa:£Xˆ0ˆ{ŒÀ•ŠòÀŽ+98†;†ÇH! † ††'†<†"!Ä@ ”"†4'W–2g•†€†*† )Ž3%†$Ž)#†(††4†††,††ŽkŽ-ˆˆ9ˆ5ˆMˆ9ˆŒˆ¦A0–E.†-ŽM††††ŠŸ†† † † † † ††Žõˆ…!‘[ŽŠExploration and ApproximŠ:Ž. L-play the†$ŽŠ.collectŠdata forŠ'point HŠSˆ7 distance GHŽ‚R-drag the x-coordinate of H andŠlength GH back toŠ0 curve, we see a mimimum.[Ž Analytically:[ŒFLet BE=p, DE=q, EH=aŠrGF=b,ˆ’nˆ\consider two s†cla† riangles ˆmGFDŽXж† ˆ…DEHŒŽ.†zŽ{WŒ³Œ©GFŒ²’E=DEŒÎEHº or<bŒíp’:ŽWqaŒ‚. Šý‰Step 1.ŒQ †‘no‡vthat b’CpžDÆ”D2. We define the ˆsquar†istanceŒ-,+ function of GH below by using Pythagorean Œ_The†m.RˆmŒs Clear_a_z R†nŒ‰done–&Ž•f(a,b)=(p+a)^2+(q+b†¶9[Ž[ŽÍStep 3.ŒÅˆò substitue bˆ¿pî’qaŒ/ intoއŒÇ†W‰;b:=¨8ŠÕ¨Q—Œ×ŠÇ¶€+q«2Žy+‡¾ a+p† 2R†Œ [ˆStep 4.Œ$%E Find the critical points by settingŠderivative of f equals to 0.–hsolve(diff(f(a,b),a)=0†R†€†‡Š–a=Ž pî’q’´Ž¿1ŽÇ3ŒÏ,a=-pŠk!ŒãŒcŽ)aø|a=¬e¤dŠžŸ6î’p“IQ†¨”ŽÉ+4î’œÂãî’Ò‡ +2î’† pî’qŒ 2Œ4Œ3Ð)3R†[ŒT[†6ˆ`Note.Žl&$ Since p,q>0, it is not hard to see !Œ˜f(a,b)Œ¥a¡ˆX†Ä|a=¬Ã’›1ÄŒê>0 so f indeed†us a Q minimum at â[.[’úŒø Numerically:Œþ Suppos‡ˆø4 and q‰6:—= p:=4RM† “Zq:=6R† †Œa:=Ž pî’qŽ'2Ž11Ž93ŠFŽ92î’18¨$†Ž bŠ1Ž#12Žt°8Œ£f(a,b)е"Š( 14.046965@F–GXD†5–l[ŒÞ,Conclusion: The above is t† minimum length.†4‰ DiscusŠ4ŽGFWhenŠ3GH reaches itsB, willŠT triangle B†+b†qn†noceles?Ž•ˆ±‡(taníü((y+q)/(x+p))/î)*180˜ù49.596534•–SCywR†† x+pR"Š8.545144TQDXˆ/˜/y+q–/ 10.206051ff’_Œ_[†^ Remark:ŒwH Since †~ is not same as †b which implies that† e triangle3an ŒÆ isoceles.[ŒnŽ'Exa†F e 2. Repla†tˆJwall by a circle\¸$ΈÓ=†×΋<ŠâŠí‰M 2U‰C† `4(iS‡p Y™bV )I‰|PhDvv‰ˆD‰‰ˆ‹8‡¢r† ‡« A†† † L† †Œ C˜ u˜x˜yŽ †N† >†WÅH ŽN?†iÀH ŠJ@†yÐ6–°†2…a€€"–†& 'R)5x†— —Ž;Ž‘ˆ3 ÈÀ ŒU–Br%hFTB†ÍY…ˆÆŒ1ŽNw‡†NYˆNˆéˆ…Š—B˜…9y5w˜”…wˆ‘¢…ˆdÄ’CC˜È3 C0#–È!¨È‰UÄ@ÈŽ˜‰D¢ˆÓ@mŽˆÕE‹U˜8?F†ÈÈ  † @†Ð6–°†XW`ˆ”†% ™)E2…Rˆ$ ™ƒE”q!†.`†A?† G†FÈÈ"ŒF†(†\(‚Y…ˆS phpŽ 2CŠ+†eg”I$’€Ž<!9 "`† Šy†T$(  &€0Žl'ˆ”! )ƒs'l1‰“„4!'65€€ލ8y¨A75"´D˜‚lGa‚0PSQB–ŽðW5¨`•w—eif‘iaSPŽðtI`u yÌ…a€€‘,‘™‚Ž`™…`0€89 Ž GuwPŽ% e@Ž$6qH€Ž0HfE2Ž<b™„3<yb`ŽT™ƒDŒ`"4"†ŽHPCpŽT†(†‚(†—saŽt……‘Ž€„wWŽPƒupƒpŒ°‚ytŽŒ1ˆÇŠ ch$€2)Ž€ydV<y€v0xU’cTxw†T)whuŠÇŠcS lwrv’0•DDx5&Q4x“—Lypqf€p&d” LƒD—™d…&”‡AŽü‰”60’•ˆpŽ –6“ Œ9IdPŒ$4ˆ QŒ<ƒa™ˆG†H$!3`ŒT2ƒ”c<B—†”ŽlT–DA`i#‰ql†9€EŽ&84Žœ37dŒ†H†²ÈÈ"€Š7ˆ(†Ä(‚Y…@ŒÎphŽÎ2CÎegŽæ”æI$’€ !9 "`† Î$( Ú&€:'ˆ”!F)ƒs'‘:1‰“R4!'^65€€‘:8yvA75"‚D˜‚lGa‚‘jP^SQB–W5`Ž `•w—eifpŽ$iaSPŽ0tI`u y0Ž$…a€€ ŽT‘™‚`™…`Ž$€89Ž$GuwŽT% e@Œ6qH€ŒœHfE2ލb™„3Ž`ybŽ´™ƒDŒÌ"4"†ŽHPCpŽT†(†î(†—saŽÈ……‘ŽÔ„wWƒupƒŽø‚ytŽŒ1‰3Š ch€2)Ž€ydV4y€v0xU’cLx(w†T)@whuŠÇŠcS lwrvXw•DDx5&Q@ “—PŽypqf`Ž €p&” <ƒD—™€ŽH…& ŽT‡ApŽ0‰”60Žl’•ˆ–6“09IdŽx4ˆŽ„QŒ¨ƒa™ˆ³†´$!3Žœ2ƒ”cŽ´B—†”ŽlT–DAŽi#‰qŽœ†9€EŽ&84Žœ37d†I‡9ÊÈ ŠÓ3‰"Ð6–°‡5E‚Dsa?"XG“Su7ˆ;Ž î IJB: ÿÿü†ž  †F† ‰€–J‡œ9ÈÈ"‹Q2˜~•‚DsaPf4ty4Š ††Œ† Length IJ: ÿÿü†ž$†K†LÄÈ ƒ †UI@†\Ð6–°†9y5w˜ˆ&f#5’ˆj¢nL†ÅH ŒCJ˜Cc5—p€fбU†TƒC†½ ™ ²†M†Ö9ÈÈ"ŒG1˜ŠE’ø°ø Coord of ´ù޶‡HN†w†w€‹‡Z(†r(‚Y…‰ ‡jphpŽ 2CŠ+eg”I$’€Ž<!9Ž "`† `$(  &€0Ž'ˆ”! )ƒs' Ž01‰“pŽ<4!' 65€€ŽT8y$A75"0D˜‚lGa‚0PSQB–ŽœW5¨`•w—eif„iaSPŽÌtI`u yÌ…a€€À‘™‚`™…`Žð€89ŽäGuwŽT% e@Œä6qH€8HfE2ލb™„3 yb\™ƒD"4"†ŽHPCpŽT†(‡Š(†—saŽÈ……‘ŽÔ„wW¬ƒupƒp‚yt€Ž 1ˆˆch$€2)Ž0ydV<y€v0xU’cTx Ž`w†T)0ŽlwhuŠGˆxwcS lwrv’0•DDŽœx5&Q@ލx“—PŽ´ypqf`Ž €p&” <ƒD—™Ø…&‡Aü‰”6œ’•ˆ‘–6“09IdŽx4ˆŽ„Qƒa™8$!3Žœ2ƒ”cŽ´B—†”T–DA i#‰q,†9€E,&84837d†O Å@  † † B†P†ÈÀ@†%Ð6–°†,r‡ia€˜†7Y…4 'R)5w‡†OY—†QˆXĘXAŽXˆ\R†v5Å@!’K†S†v ÈÀ˜v˜jQ1E˜w†¹ ‡9y5w˜†`†ÉT†XÅH ŠÎA–CŒ+Š­)G$wCйY–§U¢½ˆYˆŸV‰/ÄH!œ×SÜ:‰5‡P ™˜Q‘Y ™VAV‰  ‰BWˆhÆ@–NL†FŽX£›‰%ˆˆY†<ƆŠ †F†Z† ÀÈ Œ@†Ð6–°†#—)5w˜† €'R)5x†: ™uA ™†[†OÈÈ `ˆUF–C°C‚މ\†’ÄH Š›G²C… —žC]†Õ%ÊH!ŽÈŽ˜ˆÞ^†]Ä@âŽoˆ_‡ Æ@ ’ˆÃÀ’‰H—)A X‹r†…3‹ž‰‰U‰Yaª]ˆwˆb¢{ˆ•c‡K ‡K¥‘µ‡†Š—)5w˜† `ˆ[† \†d†$Ä@ +ˆc†b˜Z„Ž"ˆe†NÇ *f†d ÈÀ Œ@†sÐ6–°† š~’ €'R)5x†Y™†g†¦ÃH#Š‚E–C9y޵˜7 ˆ7˜^ˆ‰hŽÉމŽYˆŸˆ¹ˆˆ"ˆ]†^†Z†i‡#ÆH–6‰ˆÆj‡=ŘPˆk†ï†ï¤ï”ã‹a§yFl†¬!š¬‰8 kW–2g”‰fŠƒm Å@  † † [†k††n†Ęg†0[u- ™†oˆDš&ˆDp†ZÈÈ Œ@†iÐ6–°†pr%hFTB†{Y…x €'R)5x†XYˆXˆFq† !Æ@ ’\ˆIˆ\°Šr¢Çˆ©ˆ'ˆ+s¢¡ˆSŽ&t¢cˆ@W–2g•Š™€†D† CŽoV†UŽCT†A†ˆÏ–ˆˆäˆ1Ž‘]\†Z†f†c†`†F†K†L†P†Q†O†N†M†J†I†H†G†?‡»(†R† ††[†Given conditions:Ž -AB is perpe† cular to BC.Ž# *-The circleˆ+adjacentˆ&both †Band–2[ˆyKey:Œƒ R Let GJ=x>0Š0 IE=y>0, th†šIH=ˆŠI†$&JH=x. So we are minimizing JI=x+y but ŒÜ@subjecŠt†ªpoi†H lyˆ/onˆaŽ¿ : (x-r)^2+(yŠ=r^2.‹.( Clear_a_z RŽdone–&‘V Note. Conside‡7‡(triang‡*IBJ:‹Š^IB‹2“+BJš =JI† 2Ž.[Œ So (IE+EB) & +(GJ+BG) @¨T and ’X(y+r¢S+(xšކ=†y)’§RŠ5Ž£simplify(solve(ºZ ëŽZ¡ ,y))R y=rî’2†U -rŠ—K[ŽŽŒ³ŒLy:ÈM¼s x-rRˆ[Œ .We define th† istance function IJ as follows:–BŽ7f(x,r)=x+yRŽdone–o simplify(Œ/)Š,ŒŽxŒ—2Ž+r’Œ$œÁdiffŽN,xÄP-ržQ -2î’rî’x”b’ˆ–½solve(œq=0Œttx=r-žÇ†b,x=’Œ†y+ŒÎ!’I‘žý|x= 2Œ î’r+rRžr[Œ/ Note that !ŒAf(x,r)Ž3xVŽ0|x= k8Šk is positive so we n’\the distance function Ž„-achieves its minimum Ž6cr†Rcal†\int ¤~Šé.[ŒÿWe let x0=GJ and y0=IERŠÅŒÔx0:=r+¡1‡1‹/°#”J y†J simplify(Ž:rî’Œlx0+r’ -Ž )¦i2ˆî’rRŠŒ taníü(Œ y0+rŒ+x’ )RŽ$îŒM4[ŽH=This shows that† e ladder Œ rtest whenŠ angle IJBˆ¤ZŒ¯ or”4tri”7ŒÐ†van†c oceles.[†¨†KŽA Challenging Problem.\ Wˆªabouˆ°is?Ή%u‡1Ή7‰<ˆP‰@ tPb)1† `e6gY€Š ™Dd6†¼ wQuw`†D‰|v’K†r† Ž  A† 'LŽ'‰°C‰½†u† † Œ x˜ yŽ †+† v†4Å@  Ž4w†FÁH ŠI@†VÐ6–°†]i“‚“ˆ4 Y™’%!e†2Ž:ŽAˆsÈÀ  Š–B ™˜™6c)†B ™HbqXe †¶ —Rycr!bІ3ˆ¨ÄHž—zXŒ–šFWV46ŠX7@—A•WŠdwt&“€†.`F{Š\˜ó|‡9À’óH˜ó08†%&Šç…y“#7н˜žó}Š›ȨóHYT†!†çY—ˆÿ8BŒÿyX†gfŠÿ†3~ Å@  † † |†zŽˆĘ€†ÄH Š0G@†@Ð6–°†H7C† %hVq0†^ ˜<ŽC}†¢Y‚œYF˜Y’•f$H™†« Y™•1Yˆ†2žXxŽGˆaˆ ƒ†ÜÈÈ `Šš«xCQ†ü 00bPŒ žR„ŽBòB…Œ„!„¦„€3“y‰"ˆ„Q 2Ч¢Ö†´B€3“y0Q 2`ˆ ˆˆŒ†‡†"ÈÈ Š@†1Ð6–°†8U08 ˆBY™p—–€ŒNžBˆ´BY—ˆIpŽB—h–Š=¢B‰ŒB"ô„ІèÉøÆ‹†„ÈÈ"Šñˆ†ÂWIfQŽøVƒWƒ‹A‡RU)RPjHx6Šò‡vC)$46‘Ž)y—H!r`2<v"—’ ’Bb0Ž `Ži2Œ#ˆ$V’e—PŽ0D&‰ 0‡p€ŽH9—EpŽTt)ŽH ˆˆk ™†† †’6‡Ž€V(•Œ€9 !ŒPSg…”ŒŒ$ yh “hŒÈ hE„Ž€ xg@ŒàQ7uŒ˜(T Œørƒr°YBy(e8E•2ue(cw%Žà ”‘!6G€Œ`yi—@˜ä†Œ‡nÈÈ ‹_ˆÐ6–°‡„c@‹vY™†uRžB†B9ÈÀ"€ 1@†Ð6–°† 1&5e…p† ˆ‚H9C(Š ††Œ†5 î : ÿÿü†ž †z† ކa ÈÈ" Š–j a10h† Y™u‘˜ d† ™u„D†B&Š †3†BÈÈ  †¬A˜­ !PŒšŠ+ ¥ †æÄÈ Š…E˜ðV’e—HŒð)$1@c¨ð‘†CÄ@ ŒÈŽ˜ˆÞ’ŠH"”ˆoŽ[ˆ±–Dˆ*ˆHˆ6“†Ö•†‡‹WIfQ€EVƒWƒ`QU)RŠ÷Hx6pC)$€Ž 6‘Ž)y—`Ž$!r`2PŽ0vŽ<—’ Ž0’Bb0ŒTŽ<i2Œ#†lV’e—ŽHD&‰ŽT0‡pŽ„9—EŽœt)Ž„ ˆˆ³ ™†† †’6‡Ž€V(•Œ¼9 !ŒPSg…”ŒŒ$ yh “hŒø hE„ xg@(Q7uŽø(T @rƒr°YBy(e8EL•2ueXcw%Žà ”‘!6G€Xyi|˜ä†ˆ¿5Ä@  ††•† ÀÀ !PŒ ™'ˆ/WIfQƒ8† F5‰1y†F — L ™Ž:ˆ$Ž †–†bÀH ŠxB@†Ð6–°†5 "C¦?`˜zˆ—œCC˜C°šžC˜†õ ÈÀ Œþ–… ™˜™8ˆ— ™HYT†!ˆÀ†Ü X1@9†Y‰ ̉Æ™Hˆˆ`šŽ’ˆ0ˆˆ ›‡w ˪‚Œv†vY—šŽvi5–qЉN—œ†Ä@  † ††›†¢—ކ7ž†<&È 8•†Ÿ†V'ǘ–Ž †R'¢Rˆ¡†!ÄH!”˜†œ›W–2g”‰f@ 4!&18†´ ™&†ÁPº ™1W‰G6„†A ˆA¢†¥.𥣆3ÈÀ–ˆ†z†¤¢0ˆˆ4¥¢Jˆ4ˆˆdˆ<ˆ"ˆ@ˆˆ¦†Ó1šíˆ2ˆPHŽ §Œ"!’"¨†ÄH ‹D@‡¡Ð6–°‡¨RWB&ˆY™$"g ˜˜†u”“@ˆ ™†•†H†¡††œ†›†š†ˆ ˜† †Ÿ†žŽ0†—†–†¨†€†YŽˆˆˆ)ˆ†z†“†ކ†Œ†‹†І‰†ˆ†‡†††…†„†ƒ†x†}†‚Že|†w††Éˆq”Ž [†ßHere a†given conditions:Ž"-We draw two lines DF and DG.Ž%/’%!an ellipse that is tangent to FD †D.[ŒZ -We construct the tangent line aŒpoi†E, which is lying onŠ3 ellipse. T–@1ŠGinterse†b FD and DGˆ[IŠ H resp† ively.[Œ;1. † ˆG %Equation of˜rˆŠ as followsŒrŽz>6.25î’x^2+2.768î’y† 0.03124† î’y-11.39†$-†2847Š2.11=0Rˆ¾ŒÄJdef‰ k(x,y)=øXRŽVdone[&[ô2’ôL‰eùDG:R3 y=4.486îìë3î’x-4.088Š. We define it as f(x,y)Rˆ(ŽŒ=y†>–L+”LRŽ4done[Œ@[ŒI†n† ˆULˆq equation FD:Œl,†«-222.9ˆ§167.8˜¢thisˆ¤g ¤!ޤŒ†¤(žF†Ñª 4Objective:‰ want to †Md†ne shortest distance IH.˜Ù‰#Step 1.ŒÄH‰S”Alope ofŠN†Bge†bŽ ellipse a‡y g†„n poi†‚‹ƒ:‘@RˆBeq:=6.25î’x^2+2.768î’y† 0.03124† î’y-11.39†$-†2847Š2.11=0RŒSŠMŒ 2Œh4Ž+(346î’yž)125’+Œ¦ 781Œ‰’c000ŒÂ-'11Š¥J00Ž“-ŽI޼Œ"†"†H’%H1211”E*ˆê“8 impdiff(eq,x)Ry'=‘ -‡@Œï‡N†¶‡h+ŠÎy-‰H5ŽÆl+5536ˆ&Œ–r[†Uˆ -Step 2.1 We define the slope of IH to b˜above.Rˆ7Œ>Ž:m(x,y)=Ž-4î’Œa312500î’x+781î’y-284750Œ††,x+5536ˆ&ŒRŒcdoneŠtŒ±[Ž[†nˆÄŠÑ3ŽÑQ If E=(a,b) is a point onŠÊ ellipse and†ã, whereˆíŠ+tange†-l‰(‡‰)@Ž:) atŽSE. Now,‹8Š2equation‰=can‰> written asޏ y-b=‡0-a). ŽÌ:-We define the following function to representŠ$lˆ- IH:RˆŽCh(x,y)=y-b-m(a,b)*(x-a)RŽ*done–:[ŒBONotˆƒat kŒE0”wsŠœellipse. Since E is ˆ ”,†<ˆv=0.ŽW_g[† Step 4.ŒÁ J Now we †Ç d H, whichˆb‰interse between DG a†,IH or fŠùŠ : —''temp1:=solve({Œ5=0,70},{‡@}0ˆª‹cx= 6250000†00î’a 2ˆ +27680000†0î’bŒ2Œ!6+3124a†"-569493614544î’a-2919416Š?-5819268Œ^*625007007132î’a+2803724†qŽI 506385821Ž,y=Ž Š/†Bˆša’™Ž»+†™1’N ¹ 4+1401426ˆ¤Š·10977††|a-1277†7†ó+23281ˆÁ0â·R‰NKH_x:=getright(temp1[1])R’ÔŠ\“p¢ÖÅ‘+3124000000î’aî’b-569493614544†-2919416ˆ †819268*625007007132†+ +28037248Ž, 69506385821Rˆ0Œ6 H_y:=getright(temp1[2])RŒ\ ŠL†_Š˜Œ2Ž+†¶1’k’ Œ 4+1401426ˆÁŠÔ10977†ªˆå-1277† 7î’b+23281ˆÞ0øÔ[ˆ¸‰Step 5."7J Now we find I, which is the intersection between IH a†, DF or h(x,y)ŠsgŠ:‘P#temp2:=solve({h(x,y),gŠ},{† })R†1x=  12500000î’aŒ 2Œ( +5536Šb  -6147928†9+bî’Œ]6248ˆN+†912330Œ1 -4777266Œ… ˆ\604†x-ˆ&94316†† b-5044037Ž­,y†®ŒC-Ä 2786Œ»†º’¹Žâ+ˆy97†IнžÙ2+13926†¾†öˆs441331-1103†ºˆ‹1911242‡!εR‹`‹fI_x:=getright(‹¦[1]Œ€ 12500000î’a†2Œ +5536Šb  -6147928†9+bî’Œ=6248ˆN+†912330Œ1 -4777266Œe ˆ\604†x-ˆ&94316†† b-5044037R†žŒ‘I_y:=getright(temp2[2])R޶-Ž  2786ŒØ†×’ÖŽ)+ˆ–97†fŠÚžö2+13926†Û‡ˆ441331#-1103†×ˆ¨1911242‡>äÒ[†‰mStep 6.z6= Set up the distance function and subject to k(a,b)=0:Rˆ 4define Lˆ,L1)=(I_x-H_x)^2+† y† yˆ L1*(ŒB)RŽ@done–P eq1:=diff(’S,a†uŠ4Œw L1î’Œ†25î’aŒ 2Œš+78†.b’+000ŽG-ŽA1139Œ 1’ +2î’w 28037ˆA0î’aqŽ/ +12417248ˆb’ ++14014264î’aî’b-5109777ŒM-1277† 7î’b+2328116†©f*5625007007132î’a+2803724800î’b-569506385821+Ž  278Š@ˆ-aŽ 2Ž) +12339744ŠL’ Œ 2+1392679ˆwˆl441331ŒL-1103891ˆ„ 1911242000Œ 5536604Š'ˆb4316Œ®0†l37Œ¨ î’Œ6#557Œ«†Ùa–|–yÎ`‘ - ’’†pÿ‰c2+13926792î’aî’b-441331000†-1103891ˆ1911242† 5536604Š)23394316†7 b-5044037Œ(2Œ2+Œ;$56075†eŠg+1401426†J b-5109777†Œf*62500†713ˆ¤ +28037248ˆ¢b-569506385821f-Žpœ<ˆ ŠDŠ€†ö”§Œ¦+†é1’c’ÇŒ4–¯‹2”²‹3277† 7î’b+23281‡•4ÖΠ—-ŒÃC+1250000††î’a+3124b-569493614544*6ˆ27007132ˆ128037248’1 506385821Œ2-Ž œ<ŽŠ”…Œk2ŽB+2768”¦b’"Œ!6š·a º†§-2919416Ž×819268âÔ”›”ÞŒ‘J‰F6‰‰A‡;7928< 553660Š1233943†’‹h044‡GT+‘2 ’2“- ‘³‡®‡2Š +5536000î’bŒ-6147928î’a+bî’Œ56248†2a+†912330Œ1 -4777266* ˆ^604î’a-ˆ(94316ˆl-5044037’rŒˆfŽ 125†ˆŸa žþ»°»Î¹Ž°-’¬6ެް Û+2768”!b¡o6+3124000000î’aî’b-569493614544†-2919416ˆ †819268*625007007132†+ +28037248Ž, 69506385821Œ1=0Rˆ9Œ?eq2:=diff(L(a,b,L1),b)ˆ$Œf L1î’Œu78†aŒ‚ˆ0ŒŽ+’#69†bŒ¥125ŽJ-ŽD2847Œ1Šýl+†È} ŠÓ†æ‹Œ2Ž/+‡=1’ò’ Œ 4+1401426‰H‹[10977‡1‰l-1277† 7î’b+23281‰eŽÍ*5625007007132î’a+2803724800î’b-569506385821+Ž  278Š@ˆ-aŽ 2Ž) +12339744ŠL’ Œ 2+1392679ˆwˆl441331ŒL-1103891ˆ„ 1911242000Œ 5536604Š'ˆb4316Œ®0†l37Œ¨ î’ޏ ˜,†þÃîÃÎÅ¡N‡w€"14014264î’a+2483449600î’b-1277†7*625007007132Š-8037†2Œ-569506385821Œ3+Œ<!1392679Œ84† 48Ž81103891Œd 553660ˆŒ†~3394316ŠŒ5044†oY-Žc ˜†y Šœ†¯ˆÉaŒ2Ž—+1241’»’ Œ4+—ˆ£510977†ú†Šˆ˜ î’b+23281†¢00ã& —+‡W†Ã46‡Ž‡ˆa+‡#72ŠËb+92‡-‡0• *5536604î’a-1233943160î’b-5044037+Ž  28†2480†'Ž 6250000†ˆaŽ72Ž@+2768”!b’"Œ!6+3124’@Š‚ 6949361454Š 29194†žŒŸ819268Œ€*Š€7007132†-+˜¦ŠK 506385821”›Ü-Œå'šƒ+‰”Ób”€âs¥E™l“E 1E‡Ž¡  ‰<™=-6147928î’a+bî’ˆ62480ˆ†912330Œ -4777266* 5536604†A-ˆ(94316†: b-5044037ŒR2Œ[ˆfŽ 1250000ˆm’,Œ† +ˆ]Šb ô»Î¹Ž°-’¬6ެް Û+2768”!b Ñ6+3124’@‹&69493‡†54‹D29194‡BC8192689*4625007007132î’a+2803724800î’b-569506385821=0RŠŠeq3:=diff(L(a,b,L1)†)ˆ%Œ7 Šh†TaŒH2Ž +276Œo’Œ&+3124†–ˆ‹1139ŒB-2847ˆž1211000Œ™ˆ œ¦[ŽŽ–Âsolve({eq1,eq2†3},{ŒË})ŠÆŒùERROR:Overflow†K‰ Remark:[ŽèQ1) It is inevitable to switchˆ an external compu†  algebra system for further wŒ+ation†Cd†Galysis.=[C2) By switching to Maple 10, we get a = 1.784381519,b†2.123030625[ŒM 3) We dra†Hhe point Ž# –HŒ‚ –JŒ”3in††ˆAGeometry Strip above†–hich will see this ŒÎ4ma†Ç es closely †×†*hatˆËhad cojectured earlier. 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And see how we can generate many different scenarios.[Q2. We will use its dynamic†Fometry a†bCAS features to help†2†b ke conjecŒŒX%before solving problems†L alytically.[Œ…Ž[†—ˆ˜ Example 1.Œ¨N A†nce q†‡et ta†² runs parallelˆ–a buildˆrat a†ûstaˆ6of pŒ9from ŽþWthe’2 . What is ˆlengthˆ8ˆ$shortest ladder tˆ*‰jreachŒTˆLgrou‡\+over the fence toŠ wall ofŠbuilding?[Note. ˆ #1For demonstration purpose, we useŒ^ollow†Fs:[Œ9$#1. We set p=4 and q=6 respectively.Ž+J2. T†E point E inœT Geometry Strip is¼coor†¼ ate (0,0).\Œ¶Example 1=>play‹animŠÂΈç†ëΆñŠ ††‰ 3X "‡ `63H‚`† ` cv"0#† b$Bf8Š ŽD‰>v’K†r‰PŽ A† 'LŽ'‰rC˜ u˜x††yŽ †Ž ÈÈ Ž'†-4fˆ8Q)u`Ž CAbPŽH ('€Ž$SWE€ Y5‡pŽeWv0r1H0yaY“@Ž`‡Vss0Žl–%SWŽ`xrtŽ)#gŽ„'’v‘Žœ@ˆ`ŽU@e•ލqyeŽ´A””ލyŠç6UpŠ×†¨eYt"Žä&CŽ$A„†T“)“ލX(‡pŽCU!Ž$X˜1  #$ƒ‘Da3‹ˆccccP‰”‹pDƒ‹o9@B‹3™f’(a0pŽ #$€`Ž‘&f Ž$`xf3y8@Ž<yaŽ0–’@Ž0‘y48Œ`†rQ0PŒls0Œxv‚ rh’ g3i„Žb€”•HXF˜d€Œ´T6I1ŽœPUu)`G62ŽØD!5SލA—I@e–ŽÀ@gbEBeb5Gi‡` W'!Àw(—Iü‰7D†‡OÄH ‹3‰\ˆÃH!‹EB‰4Ð6–°‡x%Šf – ІU†UŽCEC††Š” ž’ †3†8ÈÈ" †J)gQP†T Y™8Cg0Œ ca„ŠgY™„C$wŠ7Y™r`ŽGŽ7aHF€ŠH˜”’TuŠu ˜H˜Š ™ ™bŠ< ™IBXpŠ$™‰e`Š¥ ™B$2TŠ ™ ‰aQŽ SUЍ))h7 QX0@Œæv8T ŒòŠë5G8Ž$qˆˆŠÁ‡tŽ<Y ŠÀ'%d:y†i5‹AX™ ?V1RŽHx’$xŽœ 8)AŽx2ˆw‹«‘p`QuaX8w €Ž V) ‘@ŽT# U0Ž$R!P#y5$H1PP`ŽHFD“<DdyT`B‘”†lA'8Ql9r'™l8(xŽ6–)Yœ5y `4x„I´3˜SBÀ3Aƒ<30—´3t5À3eI’Ø4cPŽü6$(ð8eR@ðB2S’Ø‘!E0SUp‰<bux&‘Due™xT” pP8† ‡gÈÈ € ‹p†‡y(†'EŠòY™WC &Œ `e"‹RY™—‹ŽY™0rpŒ0#1CXY™%2PŒ ƒ…#`ŒY4E0Œ$U%1Fˆ1 ˜G`2tŠ ™‘4Ž F”'QpŒG58Œ0™b!sŽ BŠT#•8( IB$!Š`w—6gŒ$ 2–ЍF‡Ѝˆ–€ŒH7w8Ž•u'ŽHc"wRŠÌEq7%Ž<GU9IŽTvFV$ Œ D‰vg„Cr‹†† Y`T‹(WPG@Œ UC€vŽ,S@‘„Œ¼QB"t0IH"‹GYEXŽ\EvX#‹dD5#`B1dV‹ @qITŽx9!%€7—x@Ž 6Uyh`Ž5DgPŽ$4Q‰ Ž03x130q’€T3'U9ˆ_ˆ`3‡ 5wH6yBv$9C&E0ŽC$ŽœH34vœUEƒ‡e1•‚$yH1œ™VŽx† †âÈÈ €Š»ˆï†ô(†'EˆþY™WC &ŠnY™`e"ŠþY™—‹Y™0rpŒ0#1CX‰9†<%2‹"Y™ƒ…#Ž<Y4ŒÎY™U%1FŠl ˜G`2tŠ ™‘4Ž F”'QŠ` ™G58Šœ ™b!sŽ B"#•8(0IB$!Ž w—6gŒ 2–ˆ#†$F‡PŒ0ˆ–€Œ<7w8Ž•u'ŽHc"wR`Œ`Eq7%Ž<GU9IŽTvFV$ Œ„ D‰vg„CrŒ<†† Y`TŒ€WPG@Ž UC€vŒhS@‘„Œ¼QB"t0IH"pŽ<GYEX0EvX#D5#`B1dVŒø@qITx9!%Œø7—x„6Uyh„5Dg’¨4Q‰Œà3x1(30q’€T3'U9x3‡ˆýŒTw(6yBv9C&E0Ž C$ŽH34v@Ž$UEƒ‡PŽ0e1•‚$yH1 ŽH™VŽT† †_ÈÈ €Šˆl†q(†'Eˆ{Y™WC &ŠnY™`e"`Œ—ŠzY™0rpŒ0#1CXжY™%2Š’Y™ƒ…#Ž<Y4ŒÎY™U%1FŠl ˜G`2tŠ ™‘4Ž F”'QŠ` ™G58Šœ ™b!sŽ B"#•8( IB$!ŠÌw—6g: 2–^F‡:ˆ–€ŒH7w8R•u'ŽHc"wR‹Eq7%vGU9I²vFV$  D‰vgŒ Cr`І† Y`TPŠ WPG@Ž UC€vŽ,S@‘„Ž$QB"t0IH"pŽ<GYEXŽ\EvX#0ŽTD5#`B1dVˆ—ˆl@qITx9!%€Ž„7—x„6Uyhް5Dg’¨4Q‰Œà3x1À30q’€T3'U9x3‡ˆýŒTwH6yBvœ9C&EÀC$üH34v“ Eƒ‡Øe1•‚‘ yH1œ™VŽð† ‡’ÈÈ €‹›‰“‹‚(†'E‹¢Y™WC &Y™`e"`Ž —@Œ0rpŒ$#1CXˆ/†0%2PŒ<ƒ…#Ž<Y4E0ŒTU%1Fˆ` ˜G`2tŠ ™‘4Ž F”'QŠ` ™G58Œ0™b!sŽ BŠT#•8( IB$!Š`w—6gŠÌ 2–ˆ×†0F‡Ѝˆ–€ŒH7w8Ž•u'ŽHc"wR‹Eq7%Ž<GU9IŽTvFV$ Œ D‰vg„Cr‹D†† Y`T‹(WPG‹XUC€vŽ,S@‘„‹ˆQB"t0IH"‹|GYEXŽ\EvX#0D5#PŽ B1dVˆˆ@qIT9!%€Ž07—x@Ž<6Uyh`ŽH5DgH4Q‰ Ž`3x1`30q’€T3'U9x3‡„5wH6yBvœ9C&EÀC$ŽÌH34vœUEƒ‡Øe1•‚$yH1œ™VŽð† ‡ÈÈ €Š»‰‡$(†'E‰.Y™WC &ŠnY™`e"ŠþY™—‹Y™0rpŒ0#1CX‰i†<%2‹jY™ƒ…#Ž<Y4ŒÎY™U%1FŠl ˜G`2tŠ ™‘4Ž F”'Qp ™G58Œ b!sŽ B0ˆ$#•8( IB$!Š0w—6gŒ$ 2–ˆS†0F‡PŒ<ˆ–€ŒH7w8Ž•u'ŽHc"wR`ŒlEq7%Ž<GU9IŽTvFV$ Œ D‰vg„CrŒ<†† Y`TŒ€WPG@Œ UC€vŒhS@‘„Œ¼QB"t0IH"‹GYEX0EvX#D5#`B1dVŒø@qITx9!%Œø7—x„6Uyh„5Dg’¨4Q‰Œà3x1(30qŒø3€€3'U9ˆ Š ‡ 5w Ž$6yBv$9C&E0Ž<C$ŽHH34v@ŽTUEƒ‡PŽ`e1•‚$yH1T™VŽx††ŽÈÈ €Šsˆ›† (†'EˆªY™WC &ŠnY™`e"`Œ—ŠzY™0rpŒ0#1CXˆå†<%2Š’Y™ƒ…#Ž<Y4ŒÎY™U%1FŠl ˜G`2tŠ ™‘4Ž F”'QŠ` ™G58Šœ ™b!sŽ B"#•8( IB$!ŠÌw—6g: 2–‚F‡:ˆ–¦7w8R•u'c"wR`Œ Eq7%PŒGU9Iˆ#†$vFV$ Œ0 D‰vgŽ<CrŒ<†† Y`TŒDWPG@Ž UC€vŒhS@‘„Ž$QB"t0IH"pŽ<GYEX0EvX#0ŽTD5#`B1dVŒ¤@qITx9!%€Ž„7—x„6Uyh„5Dg’¨4Q‰Œà3x1À30q’€T3'U9x3‡ˆýŒTwH6yBvœ9C&EÀC$üH34v“ Eƒ‡Øe1•‚‘ yH1| ™V†† ÈÈ € І†(†'Eˆ'Y™WC &Œ `e"`Œ—@Œ$0rpŒ0#1CXŠbY™%2PŒHƒ…#Ž<Y4E0Œ`U%1FŠl ˜G`2tŠ ™‘4Ž F”'QŠ` ™G58Šœ ™b!sŽ BŠT#•8( IB$!ŠÌw—6gŠÌ 2– F‡Ѝˆ–€ŒH7w8Ž•u'ŽHc"wR‹Eq7%Ž<GU9I^vFV$‹W D‰vg„Cr‹D‹‚‹pY`T‹(WPG‹XUC€vŽ,S@‘„QB"tPŽ IH"pŽGYEX`Ž$EvX#0Ž0D5#0B1dVˆGˆH@qITH9!%€Ž`7—x@Žl6UyhT5Dgx4Q‰ Ž3x130q’€T3'U9x3‡„5wH6yBvœ9C&EÀC$üH34vœUEƒ‡Øe1•‚‘ yH1œ™VŽð†‡BÈÈ €Š»‰O‡T(†'EˆY™`e"‹FY™0r‹^Y™%2‹vY™Y4ŒžY™G`2tŠ  ™F”'QŠ0 ™b!sŽ #•8(0w—6gŒ F‡PŒ7w8Ž c"wR`Œ0GU9Iˆ;†< D‰vgŽHPQUƒŒ •3p2ŒTT`gsŒ<’'ƒŠ`pxŠ0`ƒ 4 Bp5Gu Œ0'@!Š„` 5‚vŠÀ`VTxˆË`•A7”ŠÌ`FƒT€Œ$0H™Ž tP˜Ž$††QY`TŒøUC€vŒìQB"tGYEXD5#0@qIT<7—x@Žø5Dg’T3x1L3€Š˜ˆ‡L6yBv C$‘@UEƒ‡”yH1 )A43PŒ h‚5@ŒaF„ €Š$'ƒpŽ0•‘‡ˆ;†<$U’pŒHWEBŒ01CRŒ0y")…0Š`eR13ŒTsugŽ•'hŒT%"†r`ŽH1W—ˆ¨ ™ˆˆŽ†¾9ÈÀ ŠÇ2ˆ¶Ð6–°†tR˜ˆ°E3•YˆŒ žC Length: ÿÿü†žc†’ŒÄÈ ‹2G˜k‰Eˆ `e(sV3‹VžK‡lÀHCH˜®“)“(˜¨¢žôõ 1@†Ð6–°†‡E!9ˆ ™9U˜ˆ ˆ*ˆ.Ž Coord: ÿÿü† †Ž†iÅH Šrˆ†{ÁH Œ„D˜„…‡CFQ4†Yˆ’Tžƒž†ÎÄ@ ’eˆ†ä ÈÈŽa–í”@†”â¤iŽKˆæÄH"”a†èŠaˆ{ˆ‰ˆ¢‹Œð!ŽðF™t‰‡†`–”¢ðœÊb‡· Ð6–°††† U‘ƒ† ‡–†† ††4Æ@  ˆ†Žˆ"ˆ†^ÅH ’*††t ËÈ#Œp@†„‹’r†˜—˜Ž(ˆKˆ‚Ę‹ˆˆaÿÿÿÿŽ"ˆq!†‹5†‹œµ"†ÿ ÈÀ˜Š™ ywR#©ˆÅH"“‹ A–ÍŒ+† `I33334†%”Z”³$¢ÉˆnˆŸ%ŠãH!œã"Ð $y1 ™†† † Y™ˆŒ ™y1D‚vˆ$˜†&†-Ç@  0†'†C ËÀ  Š@†RÐ6–°† ”W†’ ˜†€Š((†‰ÄH Š\C–GeF„18† ”F E)†CÄH#”ŸˆYˆŸ*Š˜¹ˆoˆ†ÿ+¢ˆ×,‡ÈȘ×rE%A†s;5$’6y‹G4•Bˆu`‡W-¢vŽX.‡vÆ@"’Ž/¢ªˆ40†Œ œŒÐ6–°††Š’ –`†'† †1†4Ä@  ˆ0†/dŽ"ˆ2Œ*!š*"†3†xÅ D4ˆŽÈÈ Œ@†Þ¤5†r!šœˆn4W–2g”‰f€†6¢ˆÙˆ7¢A'†8†™ †™Œóq˜š™2 ywR#‡U ‡˜²Ža†9‹H!œ{ˆa:£Xˆ0ˆ{ŒÀ•ŠòÀŽ+98†;†ÇH! † ††'†<†"!Ä@ ”"†4'W–2g•†€†*† )Ž3%†$Ž)#†(††4†††,††ŽkŽ-ˆˆ9ˆ5ˆMˆ9ˆŒˆ¦A0–E.†-ŽM††††ŠŸ†† † † † † ††Žõˆ…!‘[ŽŠExploration and ApproximŠ:Ž. L-play the†$ŽŠ.collectŠdata forŠ'point HŠSˆ7 distance GHŽ‚R-drag the x-coordinate of H andŠlength GH back toŠ0 curve, we see a mimimum.[Ž Analytically:[ŒFLet BE=p, DE=q, EH=aŠrGF=b,ˆ’nˆ\consider two s†cla† riangles ˆmGFDŽXж† ˆ…DEHŒŽ.†zŽ{WŒ³Œ©GFŒ²’E=DEŒÎEHº or<bŒíp’:ŽWqaŒ‚. Šý‰Step 1.ŒQ †‘no‡vthat b’CpžDÆ”D2. We define the ˆsquar†istanceŒ-,+ function of GH below by using Pythagorean Œ_The†m.RˆmŒs Clear_a_z R†nŒ‰done–&Ž•f(a,b)=(p+a)^2+(q+b†¶9[Ž[ŽÍStep 3.ŒÅˆò substitue bˆ¿pî’qaŒ/ intoއŒÇ†W‰;b:=¨8ŠÕ¨Q—Œ×ŠÇ¶€+q«2Žy+‡¾ a+p† 2R†Œ [ˆStep 4.Œ$%E Find the critical points by settingŠderivative of f equals to 0.–hsolve(diff(f(a,b),a)=0†R†€†‡Š–a=Ž pî’q’´Ž¿1ŽÇ3ŒÏ,a=-pŠk!ŒãŒcŽ)aø|a=¬e¤dŠžŸ6î’p“IQ†¨”ŽÉ+4î’œÂãî’Ò‡ +2î’† pî’qŒ 2Œ4Œ3Ð)3R†[ŒT[†6ˆ`Note.Žl&$ Since p,q>0, it is not hard to see !Œ˜f(a,b)Œ¥a¡ˆX†Ä|a=¬Ã’›1ÄŒê>0 so f indeed†us a Q minimum at â[.[’úŒø Numerically:Œþ Suppos‡ˆø4 and q‰6:—= p:=4RM† “Zq:=6R† †Œa:=Ž pî’qŽ'2Ž11Ž93ŠFŽ92î’18¨$†Ž bŠ1Ž#12Žt°8Œ£f(a,b)е"Š( 14.046965@F–GXD†5–l[ŒÞ,Conclusion: The above is t† minimum length.†4‰ DiscusŠ4ŽGFWhenŠ3GH reaches itsB, willŠT triangle B†+b†qn†noceles?Ž•ˆ±‡(taníü((y+q)/(x+p))/î)*180˜ù49.596534•–SCywR†† x+pR"Š8.545144TQDXˆ/˜/y+q–/ 10.206051ff’_Œ_[†^ Remark:ŒwH Since †~ is not same as †b which implies that† e triangle3an ŒÆ isoceles.[ŒnŽ'Exa†F e 2. Repla†tˆJwall by a circle\¸$ΈÓ=†×΋<ŠâŠí‰M 2U‰C† `4(iS‡p Y™bV )I‰|PhDvv‰ˆD‰‰ˆ‹8‡¢r† ‡« A†† † L† †Œ C˜ u˜x˜yŽ †N† >†WÅH ŽN?†iÀH ŠJ@†yÐ6–°†2…a€€"–†& 'R)5x†— —Ž;Ž‘ˆ3 ÈÀ ŒU–Br%hFTB†ÍY…ˆÆŒ1ŽNw‡†NYˆNˆéˆ…Š—B˜…9y5w˜”…wˆ‘¢…ˆdÄ’CC˜È3 C0#–È!¨È‰UÄ@ÈŽ˜‰D¢ˆÓ@mŽˆÕE‹U˜8?F†ÈÈ  † @†Ð6–°†XW`ˆ”†% ™)E2…Rˆ$ ™ƒE”q!†.`†A?† G†FÈÈ"ŒF†(†\(‚Y…ˆS phpŽ 2CŠ+†eg”I$’€Ž<!9 "`† Šy†T$(  &€0Žl'ˆ”! )ƒs'l1‰“„4!'65€€ލ8y¨A75"´D˜‚lGa‚0PSQB–ŽðW5¨`•w—eif‘iaSPŽðtI`u yÌ…a€€‘,‘™‚Ž`™…`0€89 Ž GuwPŽ% e@Ž$6qH€Ž0HfE2Ž<b™„3<yb`ŽT™ƒDŒ`"4"†ŽHPCpŽT†(†‚(†—saŽt……‘Ž€„wWŽPƒupƒpŒ°‚ytŽŒ1ˆÇŠ ch$€2)Ž€ydV<y€v0xU’cTxw†T)whuŠÇŠcS lwrv’0•DDx5&Q4x“—Lypqf€p&d” LƒD—™d…&”‡AŽü‰”60’•ˆpŽ –6“ Œ9IdPŒ$4ˆ QŒ<ƒa™ˆG†H$!3`ŒT2ƒ”c<B—†”ŽlT–DA`i#‰ql†9€EŽ&84Žœ37dŒ†H†²ÈÈ"€Š7ˆ(†Ä(‚Y…@ŒÎphŽÎ2CÎegŽæ”æI$’€ !9 "`† Î$( Ú&€:'ˆ”!F)ƒs'‘:1‰“R4!'^65€€‘:8yvA75"‚D˜‚lGa‚‘jP^SQB–W5`Ž `•w—eifpŽ$iaSPŽ0tI`u y0Ž$…a€€ ŽT‘™‚`™…`Ž$€89Ž$GuwŽT% e@Œ6qH€ŒœHfE2ލb™„3Ž`ybŽ´™ƒDŒÌ"4"†ŽHPCpŽT†(†î(†—saŽÈ……‘ŽÔ„wWƒupƒŽø‚ytŽŒ1‰3Š ch€2)Ž€ydV4y€v0xU’cLx(w†T)@whuŠÇŠcS lwrvXw•DDx5&Q@ “—PŽypqf`Ž €p&” <ƒD—™€ŽH…& ŽT‡ApŽ0‰”60Žl’•ˆ–6“09IdŽx4ˆŽ„QŒ¨ƒa™ˆ³†´$!3Žœ2ƒ”cŽ´B—†”ŽlT–DAŽi#‰qŽœ†9€EŽ&84Žœ37d†I‡9ÊÈ ŠÓ3‰"Ð6–°‡5E‚Dsa?"XG“Su7ˆ;Ž î IJB: ÿÿü†ž  †F† ‰€–J‡œ9ÈÈ"‹Q2˜~•‚DsaPf4ty4Š ††Œ† Length IJ: ÿÿü†ž$†K†LÄÈ ƒ †UI@†\Ð6–°†9y5w˜ˆ&f#5’ˆj¢nL†ÅH ŒCJ˜Cc5—p€fбU†TƒC†½ ™ ²†M†Ö9ÈÈ"ŒG1˜ŠE’ø°ø Coord of ´ù޶‡HN†w†w€‹‡Z(†r(‚Y…‰ ‡jphpŽ 2CŠ+eg”I$’€Ž<!9Ž "`† `$(  &€0Ž'ˆ”! )ƒs' Ž01‰“pŽ<4!' 65€€ŽT8y$A75"0D˜‚lGa‚0PSQB–ŽœW5¨`•w—eif„iaSPŽÌtI`u yÌ…a€€À‘™‚`™…`Žð€89ŽäGuwŽT% e@Œä6qH€8HfE2ލb™„3 yb\™ƒD"4"†ŽHPCpŽT†(‡Š(†—saŽÈ……‘ŽÔ„wW¬ƒupƒp‚yt€Ž 1ˆˆch$€2)Ž0ydV<y€v0xU’cTx Ž`w†T)0ŽlwhuŠGˆxwcS lwrv’0•DDŽœx5&Q@ލx“—PŽ´ypqf`Ž €p&” <ƒD—™Ø…&‡Aü‰”6œ’•ˆ‘–6“09IdŽx4ˆŽ„Qƒa™8$!3Žœ2ƒ”cŽ´B—†”T–DA i#‰q,†9€E,&84837d†O Å@  † † B†P†ÈÀ@†%Ð6–°†,r‡ia€˜†7Y…4 'R)5w‡†OY—†QˆXĘXAŽXˆ\R†v5Å@!’K†S†v ÈÀ˜v˜jQ1E˜w†¹ ‡9y5w˜†`†ÉT†XÅH ŠÎA–CŒ+Š­)G$wCйY–§U¢½ˆYˆŸV‰/ÄH!œ×SÜ:‰5‡P ™˜Q‘Y ™VAV‰  ‰BWˆhÆ@–NL†FŽX£›‰%ˆˆY†<ƆŠ †F†Z† ÀÈ Œ@†Ð6–°†#—)5w˜† €'R)5x†: ™uA ™†[†OÈÈ `ˆUF–C°C‚މ\†’ÄH Š›G²C… —žC]†Õ%ÊH!ŽÈŽ˜ˆÞ^†]Ä@âŽoˆ_‡ Æ@ ’ˆÃÀ’‰H—)A X‹r†…3‹ž‰‰U‰Yaª]ˆwˆb¢{ˆ•c‡K ‡K¥‘µ‡†Š—)5w˜† `ˆ[† \†d†$Ä@ +ˆc†b˜Z„Ž"ˆe†NÇ *f†d ÈÀ Œ@†sÐ6–°† š~’ €'R)5x†Y™†g†¦ÃH#Š‚E–C9y޵˜7 ˆ7˜^ˆ‰hŽÉމŽYˆŸˆ¹ˆˆ"ˆ]†^†Z†i‡#ÆH–6‰ˆÆj‡=ŘPˆk†ï†ï¤ï”ã‹a§yFl†¬!š¬‰8 kW–2g”‰fŠƒm Å@  † † [†k††n†Ęg†0[u- ™†oˆDš&ˆDp†ZÈÈ Œ@†iÐ6–°†pr%hFTB†{Y…x €'R)5x†XYˆXˆFq† !Æ@ ’\ˆIˆ\°Šr¢Çˆ©ˆ'ˆ+s¢¡ˆSŽ&t¢cˆ@W–2g•Š™€†D† CŽoV†UŽCT†A†ˆÏ–ˆˆäˆ1Ž‘]\†Z†f†c†`†F†K†L†P†Q†O†N†M†J†I†H†G†?‡»(†R† ††[†Given conditions:Ž -AB is perpe† cular to BC.Ž# *-The circleˆ+adjacentˆ&both †Band–2[ˆyKey:Œƒ R Let GJ=x>0Š0 IE=y>0, th†šIH=ˆŠI†$&JH=x. So we are minimizing JI=x+y but ŒÜ@subjecŠt†ªpoi†H lyˆ/onˆaŽ¿ : (x-r)^2+(yŠ=r^2.‹.( Clear_a_z RŽdone–&‘V Note. Conside‡7‡(triang‡*IBJ:‹Š^IB‹2“+BJš =JI† 2Ž.[Œ So (IE+EB) & +(GJ+BG) @¨T and ’X(y+r¢S+(xšކ=†y)’§RŠ5Ž£simplify(solve(ºZ ëŽZ¡ ,y))R y=rî’2†U -rŠ—K[ŽŽŒ³ŒLy:ÈM¼s x-rRˆ[Œ .We define th† istance function IJ as follows:–BŽ7f(x,r)=x+yRŽdone–o simplify(Œ/)Š,ŒŽxŒ—2Ž+r’Œ$œÁdiffŽN,xÄP-ržQ -2î’rî’x”b’ˆ–½solve(œq=0Œttx=r-žÇ†b,x=’Œ†y+ŒÎ!’I‘žý|x= 2Œ î’r+rRžr[Œ/ Note that !ŒAf(x,r)Ž3xVŽ0|x= k8Šk is positive so we n’\the distance function Ž„-achieves its minimum Ž6cr†Rcal†\int ¤~Šé.[ŒÿWe let x0=GJ and y0=IERŠÅŒÔx0:=r+¡1‡1‹/°#”J y†J simplify(Ž:rî’Œlx0+r’ -Ž )¦i2ˆî’rRŠŒ taníü(Œ y0+rŒ+x’ )RŽ$îŒM4[ŽH=This shows that† e ladder Œ rtest whenŠ angle IJBˆ¤ZŒ¯ or”4tri”7ŒÐ†van†c oceles.[†¨†KŽA Challenging Problem.\ Wˆªabouˆ°is?Ή%u‡1Ή7‰<ˆP‰@ tPb)1† `e6gY€Š ™Dd6†¼ wQuw`†D‰|v’K†r† Ž  A† 'LŽ'‰°C‰½†u† † Œ x˜ yŽ †+† v†4Å@  Ž4w†FÁH ŠI@†VÐ6–°†]i“‚“ˆ4 Y™’%!e†2Ž:ŽAˆsÈÀ  Š–B ™˜™6c)†B ™HbqXe †¶ —Rycr!bІ3ˆ¨ÄHž—zXŒ–šFWV46ŠX7@—A•WŠdwt&“€†.`F{Š\˜ó|‡9À’óH˜ó08†%&Šç…y“#7н˜žó}Š›ȨóHYT†!†çY—ˆÿ8BŒÿyX†gfŠÿ†3~ Å@  † † |†zŽˆĘ€†ÄH Š0G@†@Ð6–°†H7C† %hVq0†^ ˜<ŽC}†¢Y‚œYF˜Y’•f$H™†« Y™•1Yˆ†2žXxŽGˆaˆ ƒ†ÜÈÈ `Šš«xCQ†ü 00bPŒ žR„ŽBòB…Œ„!„¦„€3“y‰"ˆ„Q 2Ч¢Ö†´B€3“y0Q 2`ˆ ˆˆŒ†‡†"ÈÈ Š@†1Ð6–°†8U08 ˆBY™p—–€ŒNžBˆ´BY—ˆIpŽB—h–Š=¢B‰ŒB"ô„ІèÉøÆ‹†„ÈÈ"Šñˆ†ÂWIfQŽøVƒWƒ‹A‡RU)RPjHx6Šò‡vC)$46‘Ž)y—H!r`2<v"—’ ’Bb0Ž `Ži2Œ#ˆ$V’e—PŽ0D&‰ 0‡p€ŽH9—EpŽTt)ŽH ˆˆk ™†† †’6‡Ž€V(•Œ€9 !ŒPSg…”ŒŒ$ yh “hŒÈ hE„Ž€ xg@ŒàQ7uŒ˜(T Œørƒr°YBy(e8E•2ue(cw%Žà ”‘!6G€Œ`yi—@˜ä†Œ‡nÈÈ ‹_ˆÐ6–°‡„c@‹vY™†uRžB†B9ÈÀ"€ 1@†Ð6–°† 1&5e…p† ˆ‚H9C(Š ††Œ†5 î : ÿÿü†ž †z† ކa ÈÈ" Š–j a10h† Y™u‘˜ d† ™u„D†B&Š †3†BÈÈ  †¬A˜­ !PŒšŠ+ ¥ †æÄÈ Š…E˜ðV’e—HŒð)$1@c¨ð‘†CÄ@ ŒÈŽ˜ˆÞ’ŠH"”ˆoŽ[ˆ±–Dˆ*ˆHˆ6“†Ö•†‡‹WIfQ€EVƒWƒ`QU)RŠ÷Hx6pC)$€Ž 6‘Ž)y—`Ž$!r`2PŽ0vŽ<—’ Ž0’Bb0ŒTŽ<i2Œ#†lV’e—ŽHD&‰ŽT0‡pŽ„9—EŽœt)Ž„ ˆˆ³ ™†† †’6‡Ž€V(•Œ¼9 !ŒPSg…”ŒŒ$ yh “hŒø hE„ xg@(Q7uŽø(T @rƒr°YBy(e8EL•2ueXcw%Žà ”‘!6G€Xyi|˜ä†ˆ¿5Ä@  ††•† ÀÀ !PŒ ™'ˆ/WIfQƒ8† F5‰1y†F — L ™Ž:ˆ$Ž †–†bÀH ŠxB@†Ð6–°†5 "C¦?`˜zˆ—œCC˜C°šžC˜†õ ÈÀ Œþ–… ™˜™8ˆ— ™HYT†!ˆÀ†Ü X1@9†Y‰ ̉Æ™Hˆˆ`šŽ’ˆ0ˆˆ ›‡w ˪‚Œv†vY—šŽvi5–qЉN—œ†Ä@  † ††›†¢—ކ7ž†<&È 8•†Ÿ†V'ǘ–Ž †R'¢Rˆ¡†!ÄH!”˜†œ›W–2g”‰f@ 4!&18†´ ™&†ÁPº ™1W‰G6„†A ˆA¢†¥.𥣆3ÈÀ–ˆ†z†¤¢0ˆˆ4¥¢Jˆ4ˆˆdˆ<ˆ"ˆ@ˆˆ¦†Ó1šíˆ2ˆPHŽ §Œ"!’"¨†ÄH ‹D@‡¡Ð6–°‡¨RWB&ˆY™$"g ˜˜†u”“@ˆ ™†•†H†¡††œ†›†š†ˆ ˜† †Ÿ†žŽ0†—†–†¨†€†YŽˆˆˆ)ˆ†z†“†ކ†Œ†‹†І‰†ˆ†‡†††…†„†ƒ†x†}†‚Že|†w††Éˆq”Ž [†ßHere a†given conditions:Ž"-We draw two lines DF and DG.Ž%/’%!an ellipse that is tangent to FD †D.[ŒZ -We construct the tangent line aŒpoi†E, which is lying onŠ3 ellipse. T–@1ŠGinterse†b FD and DGˆ[IŠ H resp† ively.[Œ;1. † ˆG %Equation of˜rˆŠ as followsŒrŽz>6.25î’x^2+2.768î’y† 0.03124† î’y-11.39†$-†2847Š2.11=0Rˆ¾ŒÄJdef‰ k(x,y)=øXRŽVdone[&[ô2’ôL‰eùDG:R3 y=4.486îìë3î’x-4.088Š. We define it as f(x,y)Rˆ(ŽŒ=y†>–L+”LRŽ4done[Œ@[ŒI†n† ˆULˆq equation FD:Œl,†«-222.9ˆ§167.8˜¢thisˆ¤g ¤!ޤŒ†¤(žF†Ñª 4Objective:‰ want to †Md†ne shortest distance IH.˜Ù‰#Step 1.ŒÄH‰S”Alope ofŠN†Bge†bŽ ellipse a‡y g†„n poi†‚‹ƒ:‘@RˆBeq:=6.25î’x^2+2.768î’y† 0.03124† î’y-11.39†$-†2847Š2.11=0RŒSŠMŒ 2Œh4Ž+(346î’yž)125’+Œ¦ 781Œ‰’c000ŒÂ-'11Š¥J00Ž“-ŽI޼Œ"†"†H’%H1211”E*ˆê“8 impdiff(eq,x)Ry'=‘ -‡@Œï‡N†¶‡h+ŠÎy-‰H5ŽÆl+5536ˆ&Œ–r[†Uˆ -Step 2.1 We define the slope of IH to b˜above.Rˆ7Œ>Ž:m(x,y)=Ž-4î’Œa312500î’x+781î’y-284750Œ††,x+5536ˆ&ŒRŒcdoneŠtŒ±[Ž[†nˆÄŠÑ3ŽÑQ If E=(a,b) is a point onŠÊ ellipse and†ã, whereˆíŠ+tange†-l‰(‡‰)@Ž:) atŽSE. Now,‹8Š2equation‰=can‰> written asޏ y-b=‡0-a). ŽÌ:-We define the following function to representŠ$lˆ- IH:RˆŽCh(x,y)=y-b-m(a,b)*(x-a)RŽ*done–:[ŒBONotˆƒat kŒE0”wsŠœellipse. Since E is ˆ ”,†<ˆv=0.ŽW_g[† Step 4.ŒÁ J Now we †Ç d H, whichˆb‰interse between DG a†,IH or fŠùŠ : —''temp1:=solve({Œ5=0,70},{‡@}0ˆª‹cx= 6250000†00î’a 2ˆ +27680000†0î’bŒ2Œ!6+3124a†"-569493614544î’a-2919416Š?-5819268Œ^*625007007132î’a+2803724†qŽI 506385821Ž,y=Ž Š/†Bˆša’™Ž»+†™1’N ¹ 4+1401426ˆ¤Š·10977††|a-1277†7†ó+23281ˆÁ0â·R‰NKH_x:=getright(temp1[1])R’ÔŠ\“p¢ÖÅ‘+3124000000î’aî’b-569493614544†-2919416ˆ †819268*625007007132†+ +28037248Ž, 69506385821Rˆ0Œ6 H_y:=getright(temp1[2])RŒ\ ŠL†_Š˜Œ2Ž+†¶1’k’ Œ 4+1401426ˆÁŠÔ10977†ªˆå-1277† 7î’b+23281ˆÞ0øÔ[ˆ¸‰Step 5."7J Now we find I, which is the intersection between IH a†, DF or h(x,y)ŠsgŠ:‘P#temp2:=solve({h(x,y),gŠ},{† })R†1x=  12500000î’aŒ 2Œ( +5536Šb  -6147928†9+bî’Œ]6248ˆN+†912330Œ1 -4777266Œ… ˆ\604†x-ˆ&94316†† b-5044037Ž­,y†®ŒC-Ä 2786Œ»†º’¹Žâ+ˆy97†IнžÙ2+13926†¾†öˆs441331-1103†ºˆ‹1911242‡!εR‹`‹fI_x:=getright(‹¦[1]Œ€ 12500000î’a†2Œ +5536Šb  -6147928†9+bî’Œ=6248ˆN+†912330Œ1 -4777266Œe ˆ\604†x-ˆ&94316†† b-5044037R†žŒ‘I_y:=getright(temp2[2])R޶-Ž  2786ŒØ†×’ÖŽ)+ˆ–97†fŠÚžö2+13926†Û‡ˆ441331#-1103†×ˆ¨1911242‡>äÒ[†‰mStep 6.z6= Set up the distance function and subject to k(a,b)=0:Rˆ 4define Lˆ,L1)=(I_x-H_x)^2+† y† yˆ L1*(ŒB)RŽ@done–P eq1:=diff(’S,a†uŠ4Œw L1î’Œ†25î’aŒ 2Œš+78†.b’+000ŽG-ŽA1139Œ 1’ +2î’w 28037ˆA0î’aqŽ/ +12417248ˆb’ ++14014264î’aî’b-5109777ŒM-1277† 7î’b+2328116†©f*5625007007132î’a+2803724800î’b-569506385821+Ž  278Š@ˆ-aŽ 2Ž) +12339744ŠL’ Œ 2+1392679ˆwˆl441331ŒL-1103891ˆ„ 1911242000Œ 5536604Š'ˆb4316Œ®0†l37Œ¨ î’Œ6#557Œ«†Ùa–|–yÎ`‘ - ’’†pÿ‰c2+13926792î’aî’b-441331000†-1103891ˆ1911242† 5536604Š)23394316†7 b-5044037Œ(2Œ2+Œ;$56075†eŠg+1401426†J b-5109777†Œf*62500†713ˆ¤ +28037248ˆ¢b-569506385821f-Žpœ<ˆ ŠDŠ€†ö”§Œ¦+†é1’c’ÇŒ4–¯‹2”²‹3277† 7î’b+23281‡•4ÖΠ—-ŒÃC+1250000††î’a+3124b-569493614544*6ˆ27007132ˆ128037248’1 506385821Œ2-Ž œ<ŽŠ”…Œk2ŽB+2768”¦b’"Œ!6š·a º†§-2919416Ž×819268âÔ”›”ÞŒ‘J‰F6‰‰A‡;7928< 553660Š1233943†’‹h044‡GT+‘2 ’2“- ‘³‡®‡2Š +5536000î’bŒ-6147928î’a+bî’Œ56248†2a+†912330Œ1 -4777266* ˆ^604î’a-ˆ(94316ˆl-5044037’rŒˆfŽ 125†ˆŸa žþ»°»Î¹Ž°-’¬6ެް Û+2768”!b¡o6+3124000000î’aî’b-569493614544†-2919416ˆ †819268*625007007132†+ +28037248Ž, 69506385821Œ1=0Rˆ9Œ?eq2:=diff(L(a,b,L1),b)ˆ$Œf L1î’Œu78†aŒ‚ˆ0ŒŽ+’#69†bŒ¥125ŽJ-ŽD2847Œ1Šýl+†È} ŠÓ†æ‹Œ2Ž/+‡=1’ò’ Œ 4+1401426‰H‹[10977‡1‰l-1277† 7î’b+23281‰eŽÍ*5625007007132î’a+2803724800î’b-569506385821+Ž  278Š@ˆ-aŽ 2Ž) +12339744ŠL’ Œ 2+1392679ˆwˆl441331ŒL-1103891ˆ„ 1911242000Œ 5536604Š'ˆb4316Œ®0†l37Œ¨ î’ޏ ˜,†þÃîÃÎÅ¡N‡w€"14014264î’a+2483449600î’b-1277†7*625007007132Š-8037†2Œ-569506385821Œ3+Œ<!1392679Œ84† 48Ž81103891Œd 553660ˆŒ†~3394316ŠŒ5044†oY-Žc ˜†y Šœ†¯ˆÉaŒ2Ž—+1241’»’ Œ4+—ˆ£510977†ú†Šˆ˜ î’b+23281†¢00ã& —+‡W†Ã46‡Ž‡ˆa+‡#72ŠËb+92‡-‡0• *5536604î’a-1233943160î’b-5044037+Ž  28†2480†'Ž 6250000†ˆaŽ72Ž@+2768”!b’"Œ!6+3124’@Š‚ 6949361454Š 29194†žŒŸ819268Œ€*Š€7007132†-+˜¦ŠK 506385821”›Ü-Œå'šƒ+‰”Ób”€âs¥E™l“E 1E‡Ž¡  ‰<™=-6147928î’a+bî’ˆ62480ˆ†912330Œ -4777266* 5536604†A-ˆ(94316†: b-5044037ŒR2Œ[ˆfŽ 1250000ˆm’,Œ† +ˆ]Šb ô»Î¹Ž°-’¬6ެް Û+2768”!b Ñ6+3124’@‹&69493‡†54‹D29194‡BC8192689*4625007007132î’a+2803724800î’b-569506385821=0RŠŠeq3:=diff(L(a,b,L1)†)ˆ%Œ7 Šh†TaŒH2Ž +276Œo’Œ&+3124†–ˆ‹1139ŒB-2847ˆž1211000Œ™ˆ œ¦[ŽŽ–Âsolve({eq1,eq2†3},{ŒË})ŠÆŒùERROR:Overflow†K‰ Remark:[ŽèQ1) It is inevitable to switchˆ an external compu†  algebra system for further wŒ+ation†Cd†Galysis.=[C2) By switching to Maple 10, we get a = 1.784381519,b†2.123030625[ŒM 3) We dra†Hhe point Ž# –HŒ‚ –JŒ”3in††ˆAGeometry Strip above†–hich will see this ŒÎ4ma†Ç es closely †×†*hatˆËhad cojectured earlier. 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œR…‘a€ €„ËX €à¬fb u4ºba Ðr@b €Ðïv˜„a *ç„‘a  €Ï9™„ r§b œR…‘a€ €Ðïv˜„ u4ºb TÎ #a € €XÏ] ôb @x}a €E÷L €xzŒIb \{Ta€ \{T €E÷L €xzŒIb \{Ta€ €2åH  Ê^7 ôab yTb €XÏ]a  ¼¾a  € €Ï9™„ r§b œR…‘a€ €„ËX €à¬fb u4ºba Ðr@b €Ðïv˜„a *ç„‘a  €E÷L €xzŒIb \{Ta€ €2åH  Ê^7 ôab yTb €XÏ]a  ¼¾a   €œR…‘ €Ï9™„ r§b œR…‘a€  ;ÄŠ €ÍßÂb 8×Õba €7Jb €h‘0a pÔ§a  €Ï9™„ r§b œR…‘a€ €h‘0 8×Õb à¨;Na €\{T €=ëq €Øb ˆÔba €ôŒIb €8-Na ͦa €E÷L €xzŒIb \{Ta€  €8-N ˆÔb  š%La €E÷L €xzŒIb \{Ta€   €=ëq €Øb ˆÔba €ôŒIb €8-Na ͦa €E÷L €xzŒIb \{Ta€  €Ï9™„ r§b œR…‘a€  ;ÄŠ €ÍßÂb 8×Õba €7Jb €h‘0a pÔ§aeqF €»d €  †y  ¨ayx Z}y €sdx x1eq1 Ø ØF   €sd  ¨ab aL1 €  €œR…‘ €Ï9™„ r§b œR…‘a€ €„ËX €à¬fb u4ºba Ðr@b €Ðïv˜„a *ç„‘a  €Ï9™„ r§b œR…‘a€ €Ðïv˜„ u4ºb TÎ #a € €XÏ] ôb @x}a €E÷L €xzŒIb \{Ta€ \{T €E÷L €xzŒIb \{Ta€ €2åH  Ê^7 ôab yTb €XÏ]a  ¼¾a  € €Ï9™„ r§b œR…‘a€ €„ËX €à¬fb u4ºba Ðr@b €Ðïv˜„a *ç„‘a  €E÷L €xzŒIb \{Ta€ €2åH  Ê^7 ôab yTb €XÏ]a  ¼¾a   €œR…‘ €Ï9™„ r§b œR…‘a€  ;ÄŠ €ÍßÂb 8×Õba €7Jb €h‘0a pÔ§a  €Ï9™„ r§b œR…‘a€ €h‘0 8×Õb à¨;Na €\{T €=ëq €Øb ˆÔba €ôŒIb €8-Na ͦa €E÷L €xzŒIb \{Ta€  €8-N ˆÔb  š%La €E÷L €xzŒIb \{Ta€   €=ëq €Øb ˆÔba €ôŒIb €8-Na ͦa €E÷L €xzŒIb \{Ta€  €Ï9™„ r§b œR…‘a€  ;ÄŠ €ÍßÂb 8×Õba €7Jb €h‘0a pÔ§aeq2 Ô ÔF   €  † ´}b  ¨aaL1   €r§ €Ï9™„ r§b œR…‘a€  ;ÄŠ €ÍßÂb 8×Õba €7Jb €h‘0a pÔ§a  €Ï9™„ r§b œR…‘a€ €Íß o”b 8×Õa xzŒI €=ëq €Øb ˆÔba €ôŒIb €8-Na ͦa €E÷L €xzŒIb \{Ta€  €E÷L €xzŒIb \{Ta€ €Ø é“b ˆÔa   €=ëq €Øb ˆÔba €ôŒIb €8-Na ͦa €E÷L €xzŒIb \{Ta€  €Ï9™„ r§b œR…‘a€  ;ÄŠ €ÍßÂb 8×Õba €7Jb €h‘0a pÔ§a   € €Ï9™„ r§b œR…‘a€ €„ËX €à¬fb u4ºba Ðr@b €Ðïv˜„a *ç„‘a  €E÷L €xzŒIb \{Ta€ €2åH  Ê^7 ôab yTb €XÏ]a  ¼¾a  r§ €Ï9™„ r§b œR…‘a€ €„ËX €à¬fb u4ºba Ðr@b €Ðïv˜„a *ç„‘a € €Ï9™„ r§b œR…‘a€ €à¬f  !å€b u4ºa xzŒI €E÷L €xzŒIb \{Ta€ €2åH  Ê^7 ôab yTb €XÏ]a  ¼¾a  €E÷L €xzŒIb \{Ta€ Ê^7 ò¨b ôaeq3ààF  † €xz € b 4 ba @9b €8aa h‰ a^fH$a,b(p+a)^2+(q+b)^2f1H$x,y(p+x)^2+(q+y)^2lineFGF  €by €  €8a € €rb 4 a€ 4  € €rb 4 a€b Ð € €rb 4 a€a  €axp q temp1 X((Fx  €Ï9™„ r§b œR…‘a€ €„ËX €à¬fb u4ºba Ðr@b €Ðïv˜„a *ç„‘a$Fy  €Ï9™„ r§b œR…‘a€  ;ÄŠ €ÍßÂb 8×Õba €7Jb 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