京都府立医科大学 前期 2013年度 問1

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大学名 京都府立医科大学
学科・方式 前期
年度 2013年度
問No 問1
学部 医学部
カテゴリ 二次関数
状態 解答 解説なし ウォッチリスト

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% \newcommand{\枠}[1]{\fboxsep1pt\fbox{ #1 }\,} %=========================================================== %=========================================================== % ■≦と≧の定義 %----------------------------------------------------------- \def\le{\leqq} \def\ge{\geqq} \newcommand{\LEQQ}{\leqq} \newcommand{\GEQQ}{\geqq} % \newcommand{\LEQQ}{\mathrel{\mathpalette\gl@align<}} % \newcommand{\GEQQ}{\mathrel{\mathpalette\gl@align>}} % \newcommand{\gl@align}[2]{\lower.6ex\vbox{\baselineskip\z@skip\lineskip\z@ % \ialign{$\m@th#1\hfil##\hfil$\crcr#2\crcr=\crcr}}} % エラーになる。理由はよく解らん。 %=========================================================== %=========================================================== % ■行列 2×2 %----------------------------------------------------------- \newcommand{\matrixTT}[4]{ \begin{pmatrix} #1 & #2 \\ #3 & #4 \end{pmatrix} } %=========================================================== %=========================================================== % ■行列 3×3 %----------------------------------------------------------- \newcommand{\matrixTTT}[9]{ \begin{pmatrix} #1 & #2 & #3\\ #4 & #5 & #6\\ #7 & #8 & #9 \end{pmatrix} } %=========================================================== %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % ■ 3*1行列の定義 \newcommand{\matrixthreeone}[3]{ \left(\begin{array}{c} #1\\ #2\\ #3 \end{array}\right) } %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %=========================================================== % ■角括弧 複数行 #1 は横幅%,#2は内容 % %----------------------------------------------------------- \newcommand{\角括弧}[2]{% $% \left[ \begin{tabular}{@{}p{.#1\linewidth}}#2\end{tabular} \right]% $% } %=========================================================== %=========================================================== % ■丸括弧 複数行 #1 は横幅%,#2は内容 %----------------------------------------------------------- \newcommand{\丸括弧}[2]{% $% \left( \begin{tabular}{@{}p{.#1\linewidth}}#2\end{tabular} \right)% $% } %=========================================================== %=========================================================== % ■丸囲み数字 ①とか %----------------------------------------------------------- \newcommand{\tyoMaru}[1]{\mbox{{\normalsize \textcircled{\raisebox{-.25ex}{#1}}}}} %=========================================================== %=========================================================== % ■ 微分 df/dt %----------------------------------------------------------- \newcommand{\dd}[2]{% \dfrac{{\rm d}#1}{{\rm d}#2}% } %=========================================================== %=========================================================== % ■ ∫記号のdisplaystyleマクロ %----------------------------------------------------------- % \newcommand{\dint}{\displaystyle\int}% emathと干渉しないように\defで定義 \def\dint{\displaystyle\int} %=========================================================== %=========================================================== % ■ Σ記号のdisplaystyleマクロ %----------------------------------------------------------- \newcommand{\dsum}{\displaystyle\sum} %=========================================================== %=========================================================== % ■ lim記号のdisplaystyleマクロ %----------------------------------------------------------- \newcommand{\dlim}{\displaystyle\lim} \newcommand{\Dsum}{\sum\limits}% ちっさいΣ %=========================================================== %=========================================================== % ■ nCr マクロ %----------------------------------------------------------- \newcommand{\nCr}[2]{% {}_{#1}\mathrm{C}_{#2}% } %=========================================================== %=========================================================== % ■ 証明終了記号 ■ %----------------------------------------------------------- \newcommand{\■}{{\tiny \text{■}}} %=========================================================== \begin{document} %%%%% ■ 本文開始 ■ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % \begin{FRAME}% この間に設問を書く $a$,$b$,$c$を定数とし,$a\neq0,\,1$とする. 座標平面上に2つの放物線 \[C:y=x^2,\quad C':y=ax^2+bx+c\] がある.$C$,$C'$の両方に接する直線を$C$,$C'$の共通接線という. $C$,$C'$の共通接線がちょうど2本存在するという条件を (T) で表す. \begin{enumerate} \item 条件 (T) が成り立つための必要十分条件は,$C'$が下に凸で $C$と$C'$が異なる2点で交わるか, または,$C'$が上に凸で$C$と$C'$が共有点をもたないことのいずれかが 成り立つことであることを証明せよ. \item 条件 (T) が成り立つとき,2本の共通接線を$l$,$m$とおく. $l$と$C$,$C'$の接点をそれぞれA,Pとおき, $m$と$C$,$C'$の接点をそれぞれB,Qとおく. ただし,Aの$x$座標はBの$x$座標より小さいとする. このとき,直線ABとPQは平行であることを証明せよ. \item 条件 (T) が成り立つとき,(2)で定めた4点A,B,P,Qを 頂点とする四角形が平行四辺形となるための放物線$C'$に関する条件を 求めよ. \end{enumerate} % \end{FRAME} %--- 解答 ------------------------------------------------------------------------ %%%%%%% ■ 本文終了 ■ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \end{document}