大阪大学 前期理系 2003年度 問5

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解答作成者: 森 宏征

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入試情報

大学名 大阪大学
学科・方式 前期理系
年度 2003年度
問No 問5
学部 理学部 ・ 医学部 ・ 歯学部 ・ 薬学部 ・ 工学部 ・ 基礎工学部
カテゴリ 図形と方程式
状態 解答 解説なし ウォッチリスト

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\documentclass[a4paper,12pt,fleqn]{jreport} \setlength{\topmargin}{-25mm} \setlength{\oddsidemargin}{2.5mm} \setlength{\textwidth}{420pt} \setlength{\textheight}{700pt} \usepackage{amsmath} \usepackage{amssymb} \usepackage{ascmac} \usepackage{graphicx} \usepackage{delarray} \usepackage{multicol} \usepackage{amscd} \usepackage{pifont} \usepackage{color} \ExecuteOptions{usename} %iwa \usepackage{vector3} \usepackage{custom_mori} \begin{document} \setlength{\abovedisplayskip}{0.5zw} \setlength{\belowdisplayskip}{0.5zw} \begin{FRAME} \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \item  平面上において座標軸に平行な主軸(長軸,短軸)をもち, $x$軸,$y$軸の両方に接する楕円を考える. その中心の$x$座標を $a$ とする. このような楕円のうち点$\A(1,\,\,2)$を通るものが存在するための $a$ の範囲を 求めよ. ただし,円は楕円の特別な場合とみなすものとする. \item  (1)の楕円がちょうど2つ存在するような $a$ に対して, その2つの楕円の中心をB,Cとする. $\triangle\A\B\C$の面積を $S(a)$ で表すときこの関数のグラフを書け. \end{enumerate} \end{FRAME} \noindent{\color[named]{BurntOrange}\bfseries \fbox{解答}} \begin{enumerate} \renewcommand{\labelenumi}{(\arabic{enumi})} \begin{minipage}{270pt} \item  条件をみたす楕円の中心は第1象限内の点である. よって $a > 0$ であり, 楕円の式は \begin{align*} \frac{(x - a)^2}{a^2} + \frac{(y - b)^2}{b^2} = 1,\quad b > 0 \end{align*} とおける. これが$\A(1,\,\,2)$を通る条件は, \begin{gather*} \frac{(1 - a)^2}{a^2} + \frac{(2 - b)^2}{b^2} = 1 \\ \therefore \,\,\, (a - 1)^2b^2 - 4a^2b^2 + 4a^2 = 0 \tag*{$\cdots\cdotssp\MARU{1}$} \end{gather*} \end{minipage} \begin{minipage}{140pt} \hspace*{1zw} %\input{osaka03s5f_zu_1}% %WinTpicVersion3.08 \unitlength 0.1in \begin{picture}( 9.7200, 17.0900)( 12.0000,-20.0500) % ELLIPSE 2 0 3 0 % 4 1686 1195 2010 1843 2010 1843 2010 1843 % \special{pn 8}% \special{ar 1686 1196 324 648 0.0000000 6.2831853}% % VECTOR 2 0 3 0 % 2 1200 1843 2172 1843 % \special{pn 8}% \special{pa 1200 1844}% \special{pa 2172 1844}% \special{fp}% \special{sh 1}% \special{pa 2172 1844}% \special{pa 2106 1824}% \special{pa 2120 1844}% \special{pa 2106 1864}% \special{pa 2172 1844}% \special{fp}% % VECTOR 2 0 3 0 % 2 1362 2005 1362 385 % \special{pn 8}% \special{pa 1362 2006}% \special{pa 1362 386}% \special{fp}% \special{sh 1}% \special{pa 1362 386}% \special{pa 1342 452}% \special{pa 1362 438}% \special{pa 1382 452}% \special{pa 1362 386}% \special{fp}% % DOT 0 0 3 0 % 1 1459 1653 % \special{pn 20}% \special{sh 1}% \special{ar 1460 1654 10 10 0 6.28318530717959E+0000}% % DOT 0 0 3 0 % 1 1686 1195 % \special{pn 20}% \special{sh 1}% \special{ar 1686 1196 10 10 0 6.28318530717959E+0000}% % STR 2 0 3 0 % 3 2099 1867 2099 1948 2 0 % $x$ \put(20.9900,-19.4800){\makebox(0,0)[lb]{$x$}}% % STR 2 0 3 0 % 3 1257 385 1257 466 2 0 % $y$ \put(12.5700,-4.6600){\makebox(0,0)[lb]{$y$}}% % STR 2 0 3 0 % 3 1467 1527 1467 1608 2 0 % {\footnotesize A} \put(14.6700,-16.0800){\makebox(0,0)[lb]{{\footnotesize A}}}% % STR 2 0 3 0 % 3 1570 1069 1570 1150 2 0 % {\scriptsize$(a,\,b)$} \put(15.7000,-11.5000){\makebox(0,0)[lb]{{\scriptsize$(a,\,b)$}}}% % LINE 2 2 3 0 % 2 1686 1195 1686 1843 % \special{pn 8}% \special{pa 1686 1196}% \special{pa 1686 1844}% \special{dt 0.045}% % LINE 2 2 3 0 % 2 1686 1195 1362 1195 % \special{pn 8}% \special{pa 1686 1196}% \special{pa 1362 1196}% \special{dt 0.045}% % STR 2 0 3 0 % 3 1654 1859 1654 1940 2 0 % {\footnotesize$a$} \put(16.5400,-19.4000){\makebox(0,0)[lb]{{\footnotesize$a$}}}% % STR 2 0 3 0 % 3 1273 1146 1273 1227 2 0 % {\footnotesize$b$} \put(12.7300,-12.2700){\makebox(0,0)[lb]{{\footnotesize$b$}}}% % STR 2 0 3 0 % 3 1380 1879 1380 1960 2 0 % {\footnotesize O} \put(13.8000,-19.6000){\makebox(0,0)[lb]{{\footnotesize O}}}% \end{picture}% \end{minipage} \vskip 0.5zw \noindent% $b$ の方程式\MARU{1}が少なくとも1つ正の解をもつような $a$ の範囲を 求めればよい. \begin{enumerate} \item[(i)] $a = 1$ のとき  \MARU{1}は $-4b + 4 = 0$ だから,$b = 1\,\,\,(正の解)$. \item[(ii)] $a \neq 1$ のとき  解と係数の関係より \[ (2解の和) = \frac{4a^2}{(a - 1)^2} > 0 \] だから\MARU{1}の2解は同符号だ. したがって, \begin{align*} \MARU{1}が2つの正の解をもつ \tag*{$\cdott(*)$} \end{align*} ような $a$ の範囲を求めればよい. \begin{align*} (*) \,\,\, \Longleftrightarrow \,\,\, \left\{ \begin{array}{crl} \smallskip & \mbox{(ii\,--\,1)} & \dfrac{1}{4}(判別式) \geqq 0 \\ \smallskip かつ & \mbox{(ii\,--\,2)} & (2解の和) > 0 \\ かつ & \mbox{(ii\,--\,3)} & (2解の積) > 0 \end{array} \right. \end{align*} (ii\,--\,1)の条件 \begin{gather*} 4a^4 - 4a^2(a - 1)^2 \geqq 0 \\ a^2 - (a - 1)^2 \geqq 0 \quad(\,\because\,\,\,両辺を4a^2\,で割った) \\ 2a - 1 \geqq 0 \displaybreak[0]\\ \therefore \,\,\, a \geqq \frac{1}{2} \end{gather*} (ii\,--\,2)の条件 $\dfrac{4a^2}{(a - 1)^2} > 0$ は必ず成り立つ. \smallskip また,(ii\,--\,3)の条件が必ず成立することは確認済みである. \end{enumerate}  以上より \begin{align*} (*) \,\,\, \Longleftrightarrow \,\,\, \textcolor{red}{ \boldsymbol{a \geqq \frac{1}{2}\,\,\,かつ\,\,\,a \neq 1} } \tag*{$\Ans$} \end{align*} \item  (1)の楕円がちょうど2つ存在するのは, \begin{align*} a > \frac{1}{2} \,\,\,かつ\,\,\,a \neq 1 \end{align*} のときである. このとき\MARU{1}の2解を $\alpha,\,\,\beta\,\,\,(0 < \alpha < \beta)$ とし, $\B(a,\,\,\alpha),\,\,\C(a,\,\,\beta)$としてかまわないから,\\ \begin{minipage}{280pt} \begin{align*} S(a) = \frac{1}{2}\zettaiti{a - 1}(\beta - \alpha) \end{align*} \MARU{1}を解の公式で解いて, \begin{gather*} \alpha = \frac{2a^2 - 2a\sqrt{\vphantom{b} 2a - 1}}{(a - 1)^2} \\[1mm] \beta = \frac{2a^2 + 2a\sqrt{\vphantom{b} 2a - 1}}{(a - 1)^2} \end{gather*} よって \begin{align*} S(a) = \frac{1}{2}\zettaiti{a - 1} \cdot \frac{4a\sqrt{\vphantom{b} 2a - 1}}{(a - 1)^2} = \frac{2a\sqrt{\vphantom{b} 2a - 1}}{\zettaiti{a - 1}} \end{align*} \end{minipage} \begin{minipage}{180pt} \vspace*{-2zw}\hspace*{-3zw} %\input{osaka03s5f_zu_2} %WinTpicVersion3.08 \unitlength 0.1in \begin{picture}( 13.4700, 17.6600)( 6.8000,-21.9500) % ELLIPSE 2 0 3 0 % 4 1481 1916 1875 2045 1875 2045 1875 2045 % \special{pn 8}% \special{ar 1482 1916 394 130 0.0000000 6.2831853}% % VECTOR 2 0 3 0 % 2 935 2043 2012 2043 % \special{pn 8}% \special{pa 936 2044}% \special{pa 2012 2044}% \special{fp}% \special{sh 1}% \special{pa 2012 2044}% \special{pa 1946 2024}% \special{pa 1960 2044}% \special{pa 1946 2064}% \special{pa 2012 2044}% \special{fp}% % VECTOR 2 0 3 0 % 2 1085 2195 1085 503 % \special{pn 8}% \special{pa 1086 2196}% \special{pa 1086 504}% \special{fp}% \special{sh 1}% \special{pa 1086 504}% \special{pa 1066 570}% \special{pa 1086 556}% \special{pa 1106 570}% \special{pa 1086 504}% \special{fp}% % ELLIPSE 2 0 3 0 % 4 1481 1354 1875 2044 1875 2044 1875 2044 % \special{pn 8}% \special{ar 1482 1354 394 690 0.0000000 6.2831853}% % DOT 0 0 3 0 % 1 1201 1827 % \special{pn 20}% \special{sh 1}% \special{ar 1202 1828 10 10 0 6.28318530717959E+0000}% % DOT 0 0 3 0 % 1 1481 1354 % \special{pn 20}% \special{sh 1}% \special{ar 1482 1354 10 10 0 6.28318530717959E+0000}% % LINE 1 0 3 0 % 2 1481 1354 1481 1916 % \special{pn 13}% \special{pa 1482 1354}% \special{pa 1482 1916}% \special{fp}% % ELLIPSE 2 0 3 0 % 4 1481 1916 1875 2045 1875 2045 1875 2045 % \special{pn 8}% \special{ar 1482 1916 394 130 0.0000000 6.2831853}% % DOT 0 0 3 0 % 1 1481 1916 % \special{pn 20}% \special{sh 1}% \special{ar 1482 1916 10 10 0 6.28318530717959E+0000}% % LINE 1 0 3 0 % 2 1481 1916 1196 1827 % \special{pn 13}% \special{pa 1482 1916}% \special{pa 1196 1828}% \special{fp}% % LINE 1 0 3 0 % 2 1481 1354 1201 1827 % \special{pn 13}% \special{pa 1482 1354}% \special{pa 1202 1828}% \special{fp}% % STR 2 0 3 0 % 3 1916 2081 1916 2157 2 0 % $x$ \put(19.1600,-21.5700){\makebox(0,0)[lb]{$x$}}% % STR 2 0 3 0 % 3 961 523 961 599 2 0 % $y$ \put(9.6100,-5.9900){\makebox(0,0)[lb]{$y$}}% % STR 2 0 3 0 % 3 1350 1219 1350 1296 2 0 % {\scriptsize $\C(a,\,\beta)$} \put(13.5000,-12.9600){\makebox(0,0)[lb]{{\scriptsize $\C(a,\,\beta)$}}}% % STR 2 0 3 0 % 3 970 2094 970 2170 2 0 % {\footnotesize O} \put(9.7000,-21.7000){\makebox(0,0)[lb]{{\footnotesize O}}}% % CIRCLE 3 0 3 0 % 4 1958 2258 1943 2842 1958 1226 980 1549 % \special{pn 4}% \special{ar 1958 2258 584 584 3.7688673 4.7123890}% % STR 2 0 3 0 % 3 1982 1634 1982 1711 2 0 % {\scriptsize $\B(a,\,\alpha)$} \put(19.8200,-17.1100){\makebox(0,0)[lb]{{\scriptsize $\B(a,\,\alpha)$}}}% % CIRCLE 3 0 3 0 % 4 2074 973 2096 2196 542 1596 781 2242 % \special{pn 4}% \special{ar 2074 974 1224 1224 2.3655619 2.7553599}% % STR 2 0 3 0 % 3 680 1324 680 1400 2 0 % {\scriptsize $\A(1,\,2)$} \put(6.8000,-14.0000){\makebox(0,0)[lb]{{\scriptsize $\A(1,\,2)$}}}% % ELLIPSE 3 0 3 0 % 4 1136 1641 1535 2208 1852 2215 1777 1113 % \special{pn 4}% \special{ar 1136 1642 400 568 5.7573447 6.2831853}% \special{ar 1136 1642 400 568 0.0000000 0.5137147}% % CIRCLE 3 0 3 0 % 4 2004 1958 1989 2542 2004 926 1027 1250 % \special{pn 4}% \special{ar 2004 1958 584 584 3.7686828 4.7123890}% % STR 2 0 3 0 % 3 2027 1327 2027 1404 2 0 % {\scriptsize $\beta-\alpha$} \put(20.2700,-14.0400){\makebox(0,0)[lb]{{\scriptsize $\beta-\alpha$}}}% % LINE 2 2 3 0 % 2 1481 1916 1481 2043 % \special{pn 8}% \special{pa 1482 1916}% \special{pa 1482 2044}% \special{dt 0.045}% % STR 2 0 3 0 % 3 1441 2100 1441 2138 2 0 % {\footnotesize $a$} \put(14.4100,-21.3800){\makebox(0,0)[lb]{{\footnotesize $a$}}}% % LINE 2 2 3 0 % 2 1201 2043 1201 1827 % \special{pn 8}% \special{pa 1202 2044}% \special{pa 1202 1828}% \special{dt 0.045}% % STR 2 0 3 0 % 3 1180 2132 1180 2170 2 0 % {\footnotesize 1} \put(11.8000,-21.7000){\makebox(0,0)[lb]{{\footnotesize 1}}}% \end{picture}% \end{minipage} \vskip 0.5zw \noindent$\boldsymbol{1^\circ}$ $a > 1$ のとき \smallskip  $S(a) = \dfrac{2a\sqrt{\vphantom{b} 2a - 1}}{a - 1}$ だから, \begin{align*} S'(a) &= 2 \cdot \frac{\left(\sqrt{\vphantom{b} 2a-1} + \dfrac{a}{\sqrt{\vphantom{b} 2a-1}} \right)\!(a - 1) - a\sqrt{\vphantom{b} 2a - 1}} {(a - 1)^2} \\[1mm] &= 2 \cdot \frac{(2a - 1 + a)(a - 1) - a(2a - 1)} {(a - 1)^2\sqrt{\vphantom{b} 2a - 1}} \\[1mm] &= 2 \cdot \frac{\left(a - \dfrac{3-\sqrt{\vphantom{b} 5}}{2} \right)\!\! \left(a - \dfrac{3+\sqrt{\vphantom{b} 5}}{2} \right)} {(a - 1)^2\sqrt{\vphantom{b} 2a - 1}} \end{align*} より $S(a)$ の増減表は,\\ \begin{minipage}{100pt} \newcommand{\tabtopsep}[1]{\vbox{\vbox to#1{}\vbox to1zw{}}} \begin{align*} \begin{array}{|c||c|c|c|c|c|} \hline\tabtopsep{0.8mm} a & (1) & \cdots & \frac{3+\sqrt{\vphantom{b} 5}}{2} & \cdots \\[0.2mm] \hline S'(a) & & - & 0 & + \\ \hline S(a) & & \searrow & 最小 & \nearrow \\ \hline \end{array} \end{align*} \end{minipage} \hspace*{-1zw} \begin{minipage}{100pt} \begin{align*} S\!\left(\frac{3 + \sqrt{\vphantom{b} 5}}{2} \right) &= \frac{(3 + \sqrt{\vphantom{b} 5}) \sqrt{\vphantom{b} 2 + \sqrt{\vphantom{b} 5}}} {\dfrac{\mathstrut 1 + \sqrt{\vphantom{b} 5}}{2}} \\ &= (1 + \sqrt{\vphantom{b} 5}) \sqrt{\vphantom{b} 2 + \sqrt{\vphantom{b} 5}} \end{align*} \end{minipage} \noindent$\boldsymbol{2^\circ}$ $\dfrac{1}{2} < a < 1$ のとき \smallskip  $S(a) = -\dfrac{2a\sqrt{\vphantom{b} 2a - 1}}{a - 1}$ だから, \begin{align*} S'(a) = -2 \cdot \frac{\left(a - \dfrac{3-\sqrt{\vphantom{b} 5}}{2} \right)\!\! \left(a - \dfrac{3+\sqrt{\vphantom{b} 5}}{2} \right)} {(a - 1)^2\sqrt{\vphantom{b} 2a - 1}} > 0 \quad \left(\because \,\,\,\dfrac{3-\sqrt{\vphantom{b} 5}}{2} < \dfrac{1}{2} \right) \end{align*} より $S(a)$ は単調増加関数である. \smallskip  $\boldsymbol{1^\circ},\,\,\boldsymbol{2^\circ}$より $S(a)$ のグラフは 下のようになる. \begin{center} %\input{osaka03s5f_zu_5} %WinTpicVersion3.08 \unitlength 0.1in \begin{picture}( 21.4200, 18.9900)( 10.0000,-23.5900) % STR 2 0 3 0 % 3 1049 618 1049 630 2 0 % $y$ \put(10.4900,-6.3000){\makebox(0,0)[lb]{$y$}}% % STR 2 0 3 0 % 3 3140 2125 3140 2137 4 3200 % $x$ \put(31.4000,-21.3700){\makebox(0,0)[rt]{$x$}}% % VECTOR 2 0 3 0 % 2 1167 2352 1167 537 % \special{pn 8}% \special{pa 1168 2352}% \special{pa 1168 538}% \special{fp}% \special{sh 1}% \special{pa 1168 538}% \special{pa 1148 604}% \special{pa 1168 590}% \special{pa 1188 604}% \special{pa 1168 538}% \special{fp}% % VECTOR 2 0 3 0 % 2 1000 2093 3142 2093 % \special{pn 8}% \special{pa 1000 2094}% \special{pa 3142 2094}% \special{fp}% \special{sh 1}% \special{pa 3142 2094}% \special{pa 3076 2074}% \special{pa 3090 2094}% \special{pa 3076 2114}% \special{pa 3142 2094}% \special{fp}% % FUNC 0 0 3 0 % 9 1000 537 3142 2352 1252 2093 1377 2093 1252 1963 1252 537 1377 2352 0 5 0 0 % (x*sqrt(x-1))/abs(1/2x-1) \special{pn 20}% \special{pa 1380 2052}% \special{pa 1386 2018}% \special{pa 1390 1990}% \special{pa 1396 1962}% \special{pa 1400 1932}% \special{pa 1406 1900}% \special{pa 1410 1864}% \special{pa 1416 1824}% \special{pa 1420 1782}% \special{pa 1426 1732}% \special{pa 1430 1674}% \special{pa 1436 1610}% \special{pa 1440 1534}% \special{pa 1446 1444}% \special{pa 1450 1336}% \special{pa 1456 1206}% \special{pa 1460 1044}% \special{pa 1466 838}% \special{pa 1470 566}% \special{pa 1470 538}% \special{ip}% \special{pa 1568 538}% \special{pa 1570 582}% \special{pa 1576 646}% \special{pa 1580 700}% \special{pa 1586 748}% \special{pa 1590 790}% \special{pa 1596 828}% \special{pa 1600 860}% \special{pa 1606 890}% \special{pa 1610 916}% \special{pa 1616 942}% \special{pa 1620 962}% \special{pa 1626 982}% \special{pa 1630 1002}% \special{pa 1636 1018}% \special{pa 1640 1034}% \special{pa 1646 1048}% \special{pa 1650 1060}% \special{pa 1656 1072}% \special{pa 1660 1084}% \special{pa 1666 1094}% \special{pa 1670 1104}% \special{pa 1676 1112}% \special{pa 1680 1120}% \special{pa 1686 1128}% \special{pa 1690 1134}% \special{pa 1696 1142}% \special{pa 1700 1148}% \special{pa 1706 1154}% \special{pa 1710 1160}% \special{pa 1716 1164}% \special{pa 1720 1168}% \special{pa 1726 1174}% \special{pa 1730 1178}% \special{pa 1736 1182}% \special{pa 1740 1186}% \special{pa 1746 1188}% \special{pa 1750 1192}% \special{pa 1756 1194}% \special{pa 1760 1198}% \special{pa 1766 1200}% \special{pa 1770 1202}% \special{pa 1776 1204}% \special{pa 1780 1206}% \special{pa 1786 1208}% \special{pa 1790 1210}% \special{pa 1796 1212}% \special{pa 1800 1214}% \special{pa 1806 1216}% \special{pa 1810 1216}% \special{pa 1816 1218}% \special{pa 1820 1220}% \special{pa 1826 1220}% \special{pa 1830 1222}% \special{pa 1836 1222}% \special{pa 1840 1222}% \special{pa 1846 1224}% \special{pa 1850 1224}% \special{pa 1856 1224}% \special{pa 1860 1226}% \special{pa 1866 1226}% \special{pa 1870 1226}% \special{pa 1876 1226}% \special{pa 1880 1226}% \special{pa 1886 1228}% \special{pa 1890 1228}% \special{pa 1896 1228}% \special{pa 1900 1228}% \special{pa 1906 1228}% \special{pa 1910 1228}% \special{pa 1916 1228}% \special{pa 1920 1228}% \special{pa 1926 1228}% \special{pa 1930 1228}% \special{pa 1936 1228}% \special{pa 1940 1226}% \special{pa 1946 1226}% \special{pa 1950 1226}% \special{pa 1956 1226}% \special{pa 1960 1226}% \special{pa 1966 1226}% \special{pa 1970 1224}% \special{pa 1976 1224}% \special{pa 1980 1224}% \special{pa 1986 1224}% \special{pa 1990 1222}% \special{pa 1996 1222}% \special{pa 2000 1222}% \special{pa 2006 1222}% \special{pa 2010 1220}% \special{pa 2016 1220}% \special{pa 2020 1220}% \special{pa 2026 1218}% \special{pa 2030 1218}% \special{pa 2036 1218}% \special{pa 2040 1216}% \special{pa 2046 1216}% \special{pa 2050 1214}% \special{pa 2056 1214}% \special{pa 2060 1214}% \special{pa 2066 1212}% \special{pa 2070 1212}% \special{pa 2076 1212}% \special{pa 2080 1210}% \special{pa 2086 1210}% \special{pa 2090 1208}% \special{pa 2096 1208}% \special{pa 2100 1206}% \special{pa 2106 1206}% \special{pa 2110 1206}% \special{pa 2116 1204}% \special{pa 2120 1204}% \special{pa 2126 1202}% \special{pa 2130 1202}% \special{pa 2136 1200}% \special{pa 2140 1200}% \special{pa 2146 1198}% \special{pa 2150 1198}% \special{pa 2156 1196}% \special{pa 2160 1196}% \special{pa 2166 1194}% \special{pa 2170 1194}% \special{pa 2176 1192}% \special{pa 2180 1192}% \special{pa 2186 1190}% \special{pa 2190 1190}% \special{pa 2196 1188}% \special{pa 2200 1188}% \special{pa 2206 1186}% \special{pa 2210 1186}% \special{pa 2216 1184}% \special{pa 2220 1184}% \special{pa 2226 1182}% \special{pa 2230 1182}% \special{pa 2236 1180}% \special{pa 2240 1178}% \special{pa 2246 1178}% \special{pa 2250 1176}% \special{pa 2256 1176}% \special{pa 2260 1174}% \special{pa 2266 1174}% \special{pa 2270 1172}% \special{pa 2276 1172}% \special{pa 2280 1170}% \special{pa 2286 1170}% \special{pa 2290 1168}% \special{pa 2296 1166}% \special{pa 2300 1166}% \special{pa 2306 1164}% \special{pa 2310 1164}% \special{pa 2316 1162}% \special{pa 2320 1162}% \special{pa 2326 1160}% \special{pa 2330 1158}% \special{pa 2336 1158}% \special{pa 2340 1156}% \special{pa 2346 1156}% \special{pa 2350 1154}% \special{pa 2356 1154}% \special{pa 2360 1152}% \special{pa 2366 1150}% \special{pa 2370 1150}% \special{pa 2376 1148}% \special{pa 2380 1148}% \special{pa 2386 1146}% \special{pa 2390 1144}% \special{pa 2396 1144}% \special{pa 2400 1142}% \special{pa 2406 1142}% \special{pa 2410 1140}% \special{pa 2416 1140}% \special{pa 2420 1138}% \special{pa 2426 1136}% \special{pa 2430 1136}% \special{pa 2436 1134}% \special{pa 2440 1134}% \special{pa 2446 1132}% \special{pa 2450 1130}% \special{pa 2456 1130}% \special{pa 2460 1128}% \special{pa 2466 1128}% \special{pa 2470 1126}% \special{pa 2476 1124}% \special{pa 2480 1124}% \special{pa 2486 1122}% \special{pa 2490 1122}% \special{pa 2496 1120}% \special{pa 2500 1118}% \special{pa 2506 1118}% \special{pa 2510 1116}% \special{pa 2516 1116}% \special{pa 2520 1114}% \special{pa 2526 1114}% \special{pa 2530 1112}% \special{pa 2536 1110}% \special{pa 2540 1110}% \special{pa 2546 1108}% \special{pa 2550 1108}% \special{pa 2556 1106}% \special{pa 2560 1104}% \special{pa 2566 1104}% \special{pa 2570 1102}% \special{pa 2576 1102}% \special{pa 2580 1100}% \special{pa 2586 1098}% \special{pa 2590 1098}% \special{pa 2596 1096}% \special{pa 2600 1096}% \special{pa 2606 1094}% \special{pa 2610 1092}% \special{pa 2616 1092}% \special{pa 2620 1090}% \special{pa 2626 1090}% \special{pa 2630 1088}% \special{pa 2636 1086}% \special{pa 2640 1086}% \special{pa 2646 1084}% \special{pa 2650 1084}% \special{pa 2656 1082}% \special{pa 2660 1080}% \special{pa 2666 1080}% \special{pa 2670 1078}% \special{pa 2676 1078}% \special{pa 2680 1076}% \special{pa 2686 1074}% \special{pa 2690 1074}% \special{pa 2696 1072}% \special{pa 2700 1072}% \special{pa 2706 1070}% \special{pa 2710 1068}% \special{pa 2716 1068}% \special{pa 2720 1066}% \special{pa 2726 1066}% \special{pa 2730 1064}% \special{pa 2736 1062}% \special{pa 2740 1062}% \special{pa 2746 1060}% \special{pa 2750 1060}% \special{pa 2756 1058}% \special{pa 2760 1056}% \special{pa 2766 1056}% \special{pa 2770 1054}% \special{pa 2776 1054}% \special{pa 2780 1052}% \special{pa 2786 1050}% \special{pa 2790 1050}% \special{pa 2796 1048}% \special{pa 2800 1048}% \special{pa 2806 1046}% \special{pa 2810 1044}% \special{pa 2816 1044}% \special{pa 2820 1042}% \special{pa 2826 1042}% \special{pa 2830 1040}% \special{pa 2836 1038}% \special{pa 2840 1038}% \special{pa 2846 1036}% \special{pa 2850 1036}% \special{pa 2856 1034}% \special{pa 2860 1034}% \special{pa 2866 1032}% \special{pa 2870 1030}% \special{pa 2876 1030}% \special{pa 2880 1028}% \special{pa 2886 1028}% \special{pa 2890 1026}% \special{pa 2896 1024}% \special{pa 2900 1024}% \special{pa 2906 1022}% \special{pa 2910 1022}% \special{pa 2916 1020}% \special{pa 2920 1018}% \special{pa 2926 1018}% \special{pa 2930 1016}% \special{pa 2936 1016}% \special{pa 2940 1014}% \special{pa 2946 1014}% \special{pa 2950 1012}% \special{pa 2956 1010}% \special{pa 2960 1010}% \special{pa 2966 1008}% \special{pa 2970 1008}% \special{pa 2976 1006}% \special{pa 2980 1004}% \special{pa 2986 1004}% \special{pa 2990 1002}% \special{pa 2996 1002}% \special{pa 3000 1000}% \special{pa 3006 1000}% \special{pa 3010 998}% \special{pa 3016 996}% \special{pa 3020 996}% \special{pa 3026 994}% \special{pa 3030 994}% \special{pa 3036 992}% \special{pa 3040 992}% \special{pa 3046 990}% \special{pa 3050 988}% \special{pa 3056 988}% \special{pa 3060 986}% \special{pa 3066 986}% \special{pa 3070 984}% \special{pa 3076 982}% \special{pa 3080 982}% \special{pa 3086 980}% \special{pa 3090 980}% \special{pa 3096 978}% \special{pa 3100 978}% \special{pa 3106 976}% \special{pa 3110 974}% \special{pa 3116 974}% \special{pa 3120 972}% \special{pa 3126 972}% \special{pa 3130 970}% \special{pa 3136 970}% \special{pa 3140 968}% \special{ip}% % FUNC 0 0 3 0 % 9 1105 537 3137 2352 1345 2093 1464 2093 1345 1963 1464 537 3137 2352 0 5 0 0 % (2x*sqrt(2x-1))/abs(x-1) \special{pn 20}% \special{pa 1406 2070}% \special{pa 1410 1998}% \special{pa 1416 1938}% \special{pa 1420 1868}% \special{pa 1426 1780}% \special{pa 1430 1668}% \special{pa 1436 1516}% \special{pa 1440 1298}% \special{pa 1446 964}% \special{pa 1450 538}% \special{ip}% \special{pa 1496 538}% \special{pa 1496 542}% \special{pa 1500 676}% \special{pa 1506 774}% \special{pa 1510 852}% \special{pa 1516 912}% \special{pa 1520 962}% \special{pa 1526 1002}% \special{pa 1530 1036}% \special{pa 1536 1064}% \special{pa 1540 1086}% \special{pa 1546 1106}% \special{pa 1550 1124}% \special{pa 1556 1140}% \special{pa 1560 1152}% \special{pa 1566 1164}% \special{pa 1570 1174}% \special{pa 1576 1182}% \special{pa 1580 1188}% \special{pa 1586 1196}% \special{pa 1590 1200}% \special{pa 1596 1206}% \special{pa 1600 1210}% \special{pa 1606 1214}% \special{pa 1610 1216}% \special{pa 1616 1220}% \special{pa 1620 1222}% \special{pa 1626 1224}% \special{pa 1630 1224}% \special{pa 1636 1226}% \special{pa 1640 1226}% \special{pa 1646 1228}% \special{pa 1650 1228}% \special{pa 1656 1228}% \special{pa 1660 1228}% \special{pa 1666 1228}% \special{pa 1670 1228}% \special{pa 1676 1226}% \special{pa 1680 1226}% \special{pa 1686 1226}% \special{pa 1690 1224}% \special{pa 1696 1224}% \special{pa 1700 1222}% \special{pa 1706 1220}% \special{pa 1710 1220}% \special{pa 1716 1218}% \special{pa 1720 1216}% \special{pa 1726 1214}% \special{pa 1730 1214}% \special{pa 1736 1212}% \special{pa 1740 1210}% \special{pa 1746 1208}% \special{pa 1750 1206}% \special{pa 1756 1204}% \special{pa 1760 1202}% \special{pa 1766 1200}% \special{pa 1770 1198}% \special{pa 1776 1196}% \special{pa 1780 1194}% \special{pa 1786 1192}% \special{pa 1790 1190}% \special{pa 1796 1188}% \special{pa 1800 1186}% \special{pa 1806 1184}% \special{pa 1810 1182}% \special{pa 1816 1180}% \special{pa 1820 1176}% \special{pa 1826 1174}% \special{pa 1830 1172}% \special{pa 1836 1170}% \special{pa 1840 1168}% \special{pa 1846 1166}% \special{pa 1850 1162}% \special{pa 1856 1160}% \special{pa 1860 1158}% \special{pa 1866 1156}% \special{pa 1870 1154}% \special{pa 1876 1150}% \special{pa 1880 1148}% \special{pa 1886 1146}% \special{pa 1890 1144}% \special{pa 1896 1140}% \special{pa 1900 1138}% \special{pa 1906 1136}% \special{pa 1910 1134}% \special{pa 1916 1130}% \special{pa 1920 1128}% \special{pa 1926 1126}% \special{pa 1930 1124}% \special{pa 1936 1122}% \special{pa 1940 1118}% \special{pa 1946 1116}% \special{pa 1950 1114}% \special{pa 1956 1110}% \special{pa 1960 1108}% \special{pa 1966 1106}% \special{pa 1970 1104}% \special{pa 1976 1100}% \special{pa 1980 1098}% \special{pa 1986 1096}% \special{pa 1990 1094}% \special{pa 1996 1090}% \special{pa 2000 1088}% \special{pa 2006 1086}% \special{pa 2010 1084}% \special{pa 2016 1080}% \special{pa 2020 1078}% \special{pa 2026 1076}% \special{pa 2030 1074}% \special{pa 2036 1070}% \special{pa 2040 1068}% \special{pa 2046 1066}% \special{pa 2050 1064}% \special{pa 2056 1060}% \special{pa 2060 1058}% \special{pa 2066 1056}% \special{pa 2070 1054}% \special{pa 2076 1050}% \special{pa 2080 1048}% \special{pa 2086 1046}% \special{pa 2090 1044}% \special{pa 2096 1040}% \special{pa 2100 1038}% \special{pa 2106 1036}% \special{pa 2110 1034}% \special{pa 2116 1030}% \special{pa 2120 1028}% \special{pa 2126 1026}% \special{pa 2130 1024}% \special{pa 2136 1020}% \special{pa 2140 1018}% \special{pa 2146 1016}% \special{pa 2150 1014}% \special{pa 2156 1012}% \special{pa 2160 1008}% \special{pa 2166 1006}% \special{pa 2170 1004}% \special{pa 2176 1002}% \special{pa 2180 998}% \special{pa 2186 996}% \special{pa 2190 994}% \special{pa 2196 992}% \special{pa 2200 990}% \special{pa 2206 986}% \special{pa 2210 984}% \special{pa 2216 982}% \special{pa 2220 980}% \special{pa 2226 978}% \special{pa 2230 974}% \special{pa 2236 972}% \special{pa 2240 970}% \special{pa 2246 968}% \special{pa 2250 966}% \special{pa 2256 962}% \special{pa 2260 960}% \special{pa 2266 958}% \special{pa 2270 956}% \special{pa 2276 954}% \special{pa 2280 950}% \special{pa 2286 948}% \special{pa 2290 946}% \special{pa 2296 944}% \special{pa 2300 942}% \special{pa 2306 938}% \special{pa 2310 936}% \special{pa 2316 934}% \special{pa 2320 932}% \special{pa 2326 930}% \special{pa 2330 928}% \special{pa 2336 924}% \special{pa 2340 922}% \special{pa 2346 920}% \special{pa 2350 918}% \special{pa 2356 916}% \special{pa 2360 914}% \special{pa 2366 910}% \special{pa 2370 908}% \special{pa 2376 906}% \special{pa 2380 904}% \special{pa 2386 902}% \special{pa 2390 900}% \special{pa 2396 898}% \special{pa 2400 894}% \special{pa 2406 892}% \special{pa 2410 890}% \special{pa 2416 888}% \special{pa 2420 886}% \special{pa 2426 884}% \special{pa 2430 882}% \special{pa 2436 878}% \special{pa 2440 876}% \special{pa 2446 874}% \special{pa 2450 872}% \special{pa 2456 870}% \special{pa 2460 868}% \special{pa 2466 866}% \special{pa 2470 864}% \special{pa 2476 860}% \special{pa 2480 858}% \special{pa 2486 856}% \special{pa 2490 854}% \special{pa 2496 852}% \special{pa 2500 850}% \special{pa 2506 848}% \special{pa 2510 846}% \special{pa 2516 844}% \special{pa 2520 840}% \special{pa 2526 838}% \special{pa 2530 836}% \special{pa 2536 834}% \special{pa 2540 832}% \special{pa 2546 830}% \special{pa 2550 828}% \special{pa 2556 826}% \special{pa 2560 824}% \special{pa 2566 822}% \special{pa 2570 818}% \special{pa 2576 816}% \special{pa 2580 814}% \special{pa 2586 812}% \special{pa 2590 810}% \special{pa 2596 808}% \special{pa 2600 806}% \special{pa 2606 804}% \special{pa 2610 802}% \special{pa 2616 800}% \special{pa 2620 798}% \special{pa 2626 796}% \special{pa 2630 794}% \special{pa 2636 790}% \special{pa 2640 788}% \special{pa 2646 786}% \special{pa 2650 784}% \special{pa 2656 782}% \special{pa 2660 780}% \special{pa 2666 778}% \special{pa 2670 776}% \special{pa 2676 774}% \special{pa 2680 772}% \special{pa 2686 770}% \special{pa 2690 768}% \special{pa 2696 766}% \special{pa 2700 764}% \special{pa 2706 762}% \special{pa 2710 760}% \special{pa 2716 758}% \special{pa 2720 756}% \special{pa 2726 754}% \special{pa 2730 750}% \special{pa 2736 748}% \special{pa 2740 746}% \special{pa 2746 744}% \special{pa 2750 742}% \special{pa 2756 740}% \special{pa 2760 738}% \special{pa 2766 736}% \special{pa 2770 734}% \special{pa 2776 732}% \special{pa 2780 730}% \special{pa 2786 728}% \special{pa 2790 726}% \special{pa 2796 724}% \special{pa 2800 722}% \special{pa 2806 720}% \special{pa 2810 718}% \special{pa 2816 716}% \special{pa 2820 714}% \special{pa 2826 712}% \special{pa 2830 710}% \special{pa 2836 708}% \special{pa 2840 706}% \special{pa 2846 704}% \special{pa 2850 702}% \special{pa 2856 700}% \special{pa 2860 698}% \special{pa 2866 696}% \special{pa 2870 694}% \special{pa 2876 692}% \special{pa 2880 690}% \special{pa 2886 688}% \special{pa 2890 686}% \special{pa 2896 684}% \special{pa 2900 682}% \special{pa 2906 680}% \special{pa 2910 678}% \special{pa 2916 676}% \special{pa 2920 674}% \special{pa 2926 672}% \special{pa 2930 670}% \special{pa 2936 668}% \special{pa 2940 666}% \special{pa 2946 664}% \special{pa 2950 662}% \special{pa 2956 660}% \special{pa 2960 658}% \special{pa 2966 656}% \special{pa 2970 654}% \special{pa 2976 652}% \special{pa 2980 650}% \special{pa 2986 648}% \special{pa 2990 646}% \special{pa 2996 644}% \special{pa 3000 642}% \special{pa 3006 640}% \special{pa 3010 638}% \special{pa 3016 638}% \special{pa 3020 636}% \special{pa 3026 634}% \special{pa 3030 632}% \special{pa 3036 630}% \special{pa 3040 628}% \special{pa 3046 626}% \special{pa 3050 624}% \special{pa 3056 622}% \special{pa 3060 620}% \special{pa 3066 618}% \special{pa 3070 616}% \special{pa 3076 614}% \special{pa 3080 612}% \special{pa 3086 610}% \special{pa 3090 608}% \special{pa 3096 606}% \special{pa 3100 604}% \special{pa 3106 602}% \special{pa 3110 600}% \special{pa 3116 598}% \special{pa 3120 598}% \special{pa 3126 596}% \special{pa 3130 594}% \special{pa 3136 592}% \special{sp}% % ELLIPSE 0 0 3 0 % 4 1210 521 1415 2309 1352 2731 2565 637 % \special{pn 20}% \special{ar 1210 522 206 1788 0.0095938 1.0593457}% % LINE 2 2 3 0 % 2 1452 2359 1452 537 % \special{pn 8}% \special{pa 1452 2360}% \special{pa 1452 538}% \special{dt 0.045}% % CIRCLE 2 0 3 0 % 4 1304 2095 1304 2121 1304 2121 1304 2121 % \special{pn 8}% \special{ar 1304 2096 26 26 0.0000000 6.2831853}% % STR 2 0 3 0 % 3 1050 2127 1050 2217 2 0 % {\footnotesize O} \put(10.5000,-22.1700){\makebox(0,0)[lb]{{\footnotesize O}}}% % STR 2 0 3 0 % 3 1260 2237 1260 2327 2 0 % {\footnotesize$\frac{1}{2}$} \put(12.6000,-23.2700){\makebox(0,0)[lb]{{\footnotesize$\frac{1}{2}$}}}% % STR 2 0 3 0 % 3 1480 2150 1480 2240 2 0 % {\footnotesize 1} \put(14.8000,-22.4000){\makebox(0,0)[lb]{{\footnotesize 1}}}% % DOT 0 0 3 0 % 1 1655 1229 % \special{pn 20}% \special{sh 1}% \special{ar 1656 1230 10 10 0 6.28318530717959E+0000}% % LINE 2 2 3 0 % 2 1655 1229 1655 2093 % \special{pn 8}% \special{pa 1656 1230}% \special{pa 1656 2094}% \special{dt 0.045}% % LINE 2 2 3 0 % 2 1655 1229 1165 1229 % \special{pn 8}% \special{pa 1656 1230}% \special{pa 1166 1230}% \special{dt 0.045}% % STR 2 0 3 0 % 3 1620 2150 1620 2240 2 0 % {\footnotesize $\gamma$} \put(16.2000,-22.4000){\makebox(0,0)[lb]{{\footnotesize $\gamma$}}}% % STR 2 0 3 0 % 3 1060 1170 1060 1260 2 0 % {\footnotesize $\delta$} \put(10.6000,-12.6000){\makebox(0,0)[lb]{{\footnotesize $\delta$}}}% % STR 2 0 3 0 % 3 1860 1450 1860 1550 2 0 % {\scriptsize$\gamma = \dfrac{3+\sqrt{5}}{2}$} \put(18.6000,-15.5000){\makebox(0,0)[lb]{{\scriptsize$\gamma = \dfrac{3+\sqrt{5}}{2}$}}}% % STR 2 0 3 0 % 3 1860 1650 1860 1750 2 0 % {\scriptsize$\delta = (1+\sqrt{5}\,)\sqrt{2+\sqrt{5}}$} \put(18.6000,-17.5000){\makebox(0,0)[lb]{{\scriptsize$\delta = (1+\sqrt{5}\,)\sqrt{2+\sqrt{5}}$}}}% \end{picture}% \end{center} \end{enumerate} \end{document}