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\theoremstyle{definition}
\newtheorem{problem}{Problem}

\newtheorem*{sol}{\c C\"oz\"um}
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\begin{document}

\title{Analitik Geometri (MAT 104)\\
2.\ Ara S\i nav\i%\color{blue}\c C\"oz\"umleri
}
\date{8 May\i s 2014}
\author{David Pierce}
%\abstract{}
\maketitle
\thispagestyle{empty}

\begin{problem}
  \begin{enumerate}[(a)]
  \item 
Tan\i m\i m\i za g\"ore ne zaman $a:b::c:d$\,?  
Do\u gru \c sekil \c cizmek yeterlidir.
%Anlatman\i z i\c cin \c sekil kullanabilirsiniz.
\item
Sadece bu tan\i m\i\ 
(ve \"Oklid'den bildi\u gimiz geometriyi) kullanarak
$a:b::c:d$ ise $a:b::a+c:b+d$ orant\i s\i n\i\ kan\i tlay\i n.
(Yine do\u gru \c sekil \c cizmek yeterlidir.)
%(\"Oklid'in \"onermelerinin numaralar\i n\i\ yazman\i z gerekmez.)
  \end{enumerate}
\end{problem}

\begin{solution}
  \begin{asparaenum}[(a)]
\item    
\psset{linecolor=blue,unit=4mm,labelsep=3pt}
  \begin{pspicture}(0,-1)(5,3)
    \pspolygon(0,0)(4,0)(4,3)
\psline(3.5,0)(3.5,0.5)(4,0.5)
\uput[d](2,0){$b$}
\uput[r](4,1.5){$a$}
\psarc(0,0){1}0{36.86}
%\uput{12pt}[18.4](0,0){$\theta$}
  \end{pspicture}
\psset{unit=5mm}
  \begin{pspicture}(0,-0.8)(4.8,3)
    \pspolygon(0,0)(4,0)(4,3)
\psline(3.6,0)(3.6,0.4)(4,0.4)
\uput[d](2,0){$d$}
\uput[r](4,1.5){$c$}
\psarc(0,0){0.8}0{36.86}
%\uput{12pt}[18.4](0,0){$\theta$}
  \end{pspicture}
\item
\psset{unit=10mm}
  \begin{pspicture}(-1,-0.4)(3.6,2.7)
    \pspolygon(0,0)(3.6,0)(3.6,2.7)
\psline(1.6,0)(1.6,1.2)(3.6,1.2)
\psline(3.4,0)(3.4,0.2)(3.6,0.2)
\psline(1.4,0)(1.4,0.2)(1.6,0.2)
\uput[r](3.6,0.6){$a$}
\uput[r](1.6,0.6){$a$}
\uput[r](3.6,1.95){$c$}
\uput[d](0.8,0){$b$}
\uput[d](2.6,0){$d$}
\uput[d](2.6,1.2){$d$}
\psarc(0,0){0.4}0{36.86}
\psarc(1.6,1.2){0.4}0{36.86}
  \end{pspicture}
  \end{asparaenum}
\end{solution}

\newpage
\begin{problem}
$xy$ eksenleri dik olsun
ve birim uzunlu\u gu se\c cilmi\c s olsun.
  \begin{enumerate}[(a)]
  \item 
Bir $(s,t)$ noktas\i n\i n $y=x$ do\u grusuna uzakl\i\u g\i n\i\ bulun.
\item
Bir $(s,t)$ noktas\i n\i n $x+y=1/2$ do\u grusuna uzakl\i\u g\i n\i\ bulun.
\item
Ekseni $y=x$ do\u grusu olan, k\"o\c sesi $(1/4,1/4)$ olan,
dikey kenar\i\ $\sqrt2$ olan,
ve oda\u g\i n\i n koordinatlar\i\ $1/4$'den b\"uy\"uk olan
parabol\"un denklemini bulun.
Bu denklemi,
\begin{equation*}
ax^2+by^2+cxy+dx+ey+1=0
\end{equation*}
bi\c ciminde yaz\i n.
  \end{enumerate}
\end{problem}

\begin{solution}
  \begin{asparaenum}[(a)]
    \item
$\abs{s-t}/\sqrt2$.
\item
$\abs{s+t-1/2}/\sqrt2$.
\item
$  \begin{gathered}[t]
    \left(\frac{x-y}{\sqrt2}\right)^2=\sqrt2\cdot\frac{x+y-1/2}{\sqrt2},\\
(x-y)^2=2x+2y-1,\\
x^2+y^2-2xy-2x-2y+1=0.
  \end{gathered}$
  \end{asparaenum}
\end{solution}

\newpage

\begin{problem}
  Dik $xy$ eksenlerine g\"ore,
birim uzunlu\u gunun se\c cildi\u gi durumda,
tabloyu doldurun
ve koni kesitlerini \c cizin.
\begin{center}\renewcommand{\arraystretch}{1.5}
  \begin{tabular}{r|c|c}
    &$16x^2+9y^2+256=160x$&$x^2+8x+8y=0$\\\hline
ad&\begin{solution}elips\end{solution}
&\begin{solution}parabol\end{solution}
\\\hline
k\"o\c se(ler)&\begin{solution}$(5,4)$, $(5,-4)$\end{solution}
&\begin{solution}$(-4,2)$\end{solution}
\\\hline 
odak(lar)&\begin{solution}$(5,\sqrt7)$, $(5,-\sqrt7)$\end{solution}
&\begin{solution}$(-4,0)$\end{solution}
\\\hline
eksen&\begin{solution}$x=5$\end{solution}
&\begin{solution}$x+4=0$\end{solution}
\\\hline
  \end{tabular}
\end{center}
\end{problem}

\begin{solution}
  $\begin{gathered}[t]
  \begin{aligned}[t]
&\phantom{{}\iff{}}16x^2+9y^2+256=160x\\
&\iff 16(x^2-10x+25)+9y^2+256=400\\
&\iff 16(x-5)^2+9y^2=144\\
&\iff\frac{(x-5)^2}9+\frac{y^2}{16}=1,
  \end{aligned}\\
  \begin{aligned}
x^2+8x+8y=0
&\iff x^2+8x+16=16-8y\\
&\iff(x+4)^2=-8(y-2).
  \end{aligned}
  \end{gathered}$

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\psplot{-3}3{4 1 x x mul 9 div sub sqrt mul}
\psplot{-3}3{4 1 x x mul 9 div sub sqrt mul neg}
\psline{->}(-5,0)(4,0)
\psline{->}(-5,-4)(-5,4)
\psdots(0,2.646)(0,-4)(0,4)(0,-2.646)
\uput[d](0,2.646){$(5,\sqrt7)$}
\uput[d](0,-4){$(5,-4)$}
  \end{pspicture}
\hfill
\psset{unit=3mm}
\begin{pspicture}(-8,-8)(8,8)
\psplot{-8}8{x x mul 8 div neg}
%  \psplot{-8}0{x 8 mul neg sqrt}
%  \psplot{-8}0{x 8 mul neg sqrt neg}
\psline{->}(4,-8)(4,8)
\psline{->}(-8,-2)(8,-2)
\psline(0,-8)(0,0)
\psdots(0,0)(0,-2)
\uput[u](0,0){$(-4,2)$}
\end{pspicture}%
\begin{comment}
\begin{center}%\renewcommand{\arraystretch}{1.5}
\footnotesize
  \begin{tabular}{r|c|c}
    &$16x^2+256=9y^2+160x$&$8x+y^2+8y=0$\\\hline
ad&hiperbol&parabol\\\hline
k\"o\c se(ler)&$(-8,0)$, $(-2,0)$&$(2,-4)$\\\hline
odak(lar)&$(10,0)$, $(0,0)$&$(0,-4)$\\\hline
eksen&$y=0$&$y+4=0$\\\hline
  \end{tabular}
\end{center}
\end{comment}
\end{solution}

\newpage
\begin{problem}
\c Sekillerde
\begin{compactitem}
  \item
hem $ABC$ hem $BAC$ e\u grisi, \c cap\i\ $AD$ ve merkezi $D$ olan parabol,
\item
$BE$ ve $CF$ ordinat,
\item
$DG:DA::DA:DE$,
\item
$CH\parallel BG$,
$HK\parallel DA$,
$HL\parallel BE$ 
\end{compactitem}
olsun.
A\c sa\u g\i daki i\c saretli uzunluklar tan\i mlans\i n:
\begin{align*}
\ydp{DH}&=s,&
\ydp{HC}&=t,&
\ydp{DE}&=c,&
\ydp{GE}&=e,&
\ydp{DB}&=f,&
\ydp{BG}&=g.
\end{align*}

    \psset{unit=2cm,labelsep=3pt}
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\psline(0,0)(2.6,0)     % old diameter, slope 0
\psline(0.866,0.577)(1.155,0) % old ordinate of new vertex, slope -2
\psline(1.4,2.4)(2.6,0)     % old ordinate of point
\psline(1.4,0.933)(1.866,0)
\psline(1.4,0.933)(2.133,0.933)
\parametricplot{0}{2.4}{t t mul 1 add sqrt t 2 div sub t}
\psline(0,0)(1.4,0.933)     % new diameter, slope 2/3
\psline(1.4,2.4)(1.4,0.933) % new ordinate of point
\psline(0.866,0.577)(0.866,0)
\uput[d](1,0){$A$}
\uput[ul](0.866,0.577){$B$}
\uput[u](1.4,2.4){$C$}
\uput[d](0,0){$D$}
\uput[d](1.155,0){$E$}
\uput[d](2.6,0){$F$}
\uput[d](0.866,0){$G$}
\uput[ul](1.4,0.933){$H$}
\uput[ur](2.133,0.933){$K$}
\uput[d](1.866,0){$L$}
    \end{pspicture}
\hfill
    \begin{pspicture}(0,-0.367)(2.6,2.567)
\psline(0,0)(1,0)     % old diameter, slope 0
\parametricplot{0}{2.4}{t t mul 1 add sqrt t 2 div sub t}
\psline(0,0)(0.866,0.577)%(1.4,0.933)     % new diameter, slope 2/3
\uput[d](1,0){$B$}
\uput[ul](0.866,0.577){$A$}
\uput[u](1.4,2.4){$C$}
\uput[d](0,0){$D$}
\begin{solution}\psset{linecolor=blue}
\psline(1,0)(1,0.667)  
\psline(1,0)(2.6,0)
\psline(2.6,0)(1.4,2.4)
\psline(0.866,0.577)(2.6,1.733)
\psline(2.6,1.733)(2.6,0)
\psline(1.4,2.4)(1.4,-0.8)
\psline(1.4,-0.8)(2.6,0)
\psline(1,0)(0.75,0.5)
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\uput[dr](1.4,0.933){$F$}
\uput[ul](0.75,0.5){$G$}
\uput[r](2.6,0){$H$}
\uput[l](1.4,-0.8){$K$}
\uput[u](2.6,1.733){$L$}
\end{solution}
    \end{pspicture}

\begin{enumerate}[a)]
\item 
Sa\u gdaki \c sekli tamamlay\i n.
\item
K\"u\c c\"uk harfleri kullanarak 
$\ydp{HK}$ ve $\ydp{DF}$ y\"onl\"u do\u grular\i n\i n
 uzunluklar\i n\i\ yaz\i n.
\end{enumerate}
\end{problem}

\begin{solution}
$\ydp{HK}:\ydp{HC}::\ydp{GE}:\ydp{GB}$, dolay\i s\i yla
$\ydp{HK}=-\displaystyle\frac eg\cdot t$.

$\ydp{DF}
=\ydp{DL}+\ydp{LF}
=\ydp{DL}+\ydp{HK}
=\displaystyle\frac cf\cdot s-\displaystyle\frac eg\cdot t$.
\end{solution}
\end{document}
