% (find-angg "LATEX/2015-2-GA-P1-gab.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -recorder 2015-2-GA-P1-gab.tex"))
% (defun c () (interactive) (find-LATEXsh "lualatex 2015-2-GA-P1-gab.tex"))
% (defun d () (interactive) (find-xpdfpage "~/LATEX/2015-2-GA-P1-gab.pdf"))
% (defun e () (interactive) (find-LATEX "2015-2-GA-P1-gab.tex"))
% (defun u () (interactive) (find-latex-upload-links "2015-2-GA-P1-gab"))
% (defun z () (interactive) (find-zsh "flsfiles-tgz 2015-2-GA-P1-gab.fls 2015-2-GA-P1-gab.tgz"))
% (defun z () (interactive) (find-zsh "flsfiles-zip 2015-2-GA-P1-gab.fls 2015-2-GA-P1-gab.zip"))
% (find-xpdfpage "~/LATEX/2015-2-GA-P1-gab.pdf")
% (find-xdvipage "~/LATEX/2015-2-GA-P1-gab.dvi")
% (find-sh0 "cp -v  ~/LATEX/2015-2-GA-P1-gab.pdf /tmp/")
% (find-sh0 "cp -v  ~/LATEX/2015-2-GA-P1-gab.pdf /tmp/pen/")
%   file:///home/edrx/LATEX/2015-2-GA-P1-gab.pdf
%               file:///tmp/2015-2-GA-P1-gab.pdf
%           file:///tmp/pen/2015-2-GA-P1-gab.pdf
% http://angg.twu.net/LATEX/2015-2-GA-P1-gab.pdf
\documentclass[oneside]{book}
\usepackage[colorlinks]{hyperref} % (find-es "tex" "hyperref")
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{tikz}
%
\usepackage{edrx15}               % (find-angg "LATEX/edrx15.sty")
\input edrxaccents.tex            % (find-angg "LATEX/edrxaccents.tex")
\input edrxheadfoot.tex           % (find-dn4ex "edrxheadfoot.tex")
\input istanbuldefs               % (find-LATEX "istanbuldefs.tex")
\def\expr#1{\directlua{output(tostring(#1))}}
\def\eval#1{\directlua{#1}}
%
\begin{document}

\catcode`\^^J=10
\directlua{dednat6dir = "dednat6/"}
\directlua{dofile(dednat6dir.."dednat6.lua")}
\directlua{texfile(tex.jobname)}
\directlua{verbose()}
%\directlua{output(preamble1)}
\directlua{dofile "edrxtikz.lua"} % (find-LATEX "edrxtikz.lua")
%L V.__tostring = function (v) return format("(%.3f,%.3f)", v[1], v[2]) end

\def\pu{\directlua{pu()}}

% (find-LATEX "edrx15.sty" "psm-and-pmat")
\def\dsm #1{\left |\begin{smallmatrix}#1\end{smallmatrix}\right |}
\def\dmat#1{\left |\begin{matrix}#1\end{matrix}\right |}

{\setlength{\parindent}{0em}
\footnotesize
\par Geometria Analítica
\par PURO-UFF - 2015.2
\par Mini-gabarito da P1 - Eduardo Ochs
\par Links importantes:
\par \url{http://angg.twu.net/2015.2-GA.html} (página do curso)
\par \url{http://angg.twu.net/2015.2-GA/2015.2-GA.pdf} (quadros)
\par \url{http://angg.twu.net/LATEX/2015-2-GA-P1-gab.pdf}
\par {\tt eduardoochs@gmail.com} (meu e-mail)
}

\bsk
\bsk

% Dots, labels, vectors
%
\def\uu{\vec u}
\def\vv{\vec v}
\def\ww{\vec w}
\def\Vec#1{{\overrightarrow{#1}}}
\def\VEC#1{{\overrightarrow{(#1)}}}

\def\nm#1{\|#1\|}
\def\Reg#1{(#1)}

\def\setofxyst#1{\setofst{(x,y)∈\R^2}{#1}}
\def\setofet  #1{\setofst{#1}{t∈\R}}
\def\setofeu  #1{\setofst{#1}{u∈\R}}
\def\setofpt  #1 #2 #3 #4 {\setofet{(#1,#2)+t\VEC{#3,#4}}}
\def\setofpu  #1 #2 #3 #4 {\setofeu{(#1,#2)+u\VEC{#3,#4}}}

\def\setofek  #1{\setofst{#1}{k∈\R}}
\def\setofeth #1{\setofst{#1}{θ∈\R}}

% mygrid
%
% 2015dec20
% (find-angg ".emacs.papers" "tikz")
\tikzset{mycurve/.style=very thick}
\tikzset{axis/.style=semithick}
\tikzset{tick/.style=semithick}
\tikzset{grid/.style=gray!20,very thin}
\tikzset{anydot/.style={circle,inner sep=0pt,minimum size=1.2mm}}
\tikzset{opdot/.style={anydot, draw=black,fill=white}}
\tikzset{cldot/.style={anydot, draw=black,fill=black}}
%
\def\mygrid(#1,#2) (#3,#4){
  \clip              (#1-0.4, #2-0.4) rectangle (#3+0.4, #4+0.4);
  \draw[step=1,grid] (#1-0.2, #2-0.2) grid      (#3+0.2, #4+0.2);
  \draw[axis] (-10,0) -- (10,0);
  \draw[axis] (0,-10) -- (0,10);
  \foreach \x in {-10,...,10} \draw[tick] (\x,-0.2) -- (\x,0.2);
  \foreach \y in {-10,...,10} \draw[tick] (-0.2,\y) -- (0.2,\y);
}
\def\myaxes(#1,#2) (#3,#4){
  \clip              (#1-0.4, #2-0.4) rectangle (#3+0.4, #4+0.4);
 %\draw[step=1,grid] (#1-0.2, #2-0.2) grid      (#3+0.2, #4+0.2);
  \draw[axis] (-20,0) -- (20,0);
  \draw[axis] (0,-20) -- (0,20);
  \foreach \x in {-20,...,20} \draw[tick] (\x,-0.2) -- (\x,0.2);
  \foreach \y in {-20,...,20} \draw[tick] (-0.2,\y) -- (0.2,\y);
}

% Grid color
\tikzset{grid/.style=gray!50,very thin}

\def\tikzp#1{\mat{\begin{tikzpicture}#1\end{tikzpicture}}}

\def\mydraw        #1;{\draw [mycurve]  \expr{#1};}
\def\mydot         #1;{\node [cldot] at \expr{#1} [] {};}
\def\myldot  #1 #2 #3;{\node [cldot] at \expr{#1} [label=#2:${#3}$] {};}
\def\myseg      #1 #2;{\draw [mycurve]  \expr{#1} -- \expr{#2};}
\def\mylabel #1 #2 #3;{\node []      at \expr{#1} [label=#2:${#3}$] {};}
\def\myseggrid  #1 #2;{\draw [grid]     \expr{#1} -- \expr{#2};}

\def\e{\expr}

1)
%L A, B, C, D = v(1,-1), v(2,1), v(0,-1), v(-1,-3)
\pu
$\tikzp{[scale=0.5,auto]
    \mygrid (-2,-4) (3,2);
    \draw [mycurve]  \e{A} -- \e{B} -- \e{C} -- \e{D} -- \e{A};
    \myldot A   0 A;
    \myldot B   0 B;
    \myldot C 180 C;
    \myldot D   0 D;
  }
$

$\Vec{AB} = \VEC{1,2} = \Vec{DC}$

$\Vec{BC} = \VEC{-2,-2} = \Vec{AD}$

$\text{Área} = \dsm{1 & -2 \\ 2 & -2 \\} = 2$

\bsk
\bsk

2)
%
%L A, B, C, D = v(0,1), v(2,2), v(1,3)
%L P = v(2.5, 2.25)
%L CC = v(1.6, 1.8); CCC = v(2.2, 0.6)
%L l = Line.new(A, B-A, -1,3)
\pu
$\tikzp{[scale=0.75,auto]
    \mygrid (0,0) (4,4);
    % \draw [mycurve]  \e{A} -- \e{B} -- \e{C} -- \e{D} -- \e{A};
    \mydraw l:draw();
    \draw [mycurve] \e{A} -- \e{C} -- \e{P};
    \myldot A    0 A;
    \myldot B   270 B;
    \myldot C   180 C;
    \myldot P   270 P;
  }
$
%
\qquad
%
$\tikzp{[scale=0.75,auto]
    % \mygrid (-1,-1) (4,4);
    \mygrid (0,0) (4,4);
    \draw [mycurve]  \e{A} -- \e{C} -- \e{CC} -- \e{CCC};
    \mydraw l:draw();
    \myldot A     0 A;
    \myldot B     0 B;
    \myldot C   180 C;
    \myldot CC  180 C';
    \myldot CCC 180 C'';
  }
$


2a) $l = \setofet{(0,1)+t \VEC{2,1}} = \setofxyst{y=1+\frac{x}{2}}$

2b) 
$\begin{array}[t]{l}
 r = \setofet{A+t \Vec{AC}} = \setofxyst{y=1+2x} \\
 s = \setofet{C+t \VEC{2,1}} = \setofxyst{y=3.5-\frac{x}{2}} \\
 P = (x,y) ∈ l∩s \\
 1+\frac{x}{2} = 3.5-\frac{x}{2} \quad⇒\quad x = 2.5 \\
 y = 1+\frac{x}{2} = 1+\frac{2.5}{2} = 2.25 \quad⇒\quad P = (2.5, 2.25) \\
\end{array}
$

2c)
$\Pr_{\Vec{AB}} \Vec{PC}
 = \Pr_{\VEC{2,1}} \VEC{-1.5,0.75}
 = \frac{-3+0.75}{5} \VEC{2,1}
 = -0.45 \VEC{2,1}
 = \VEC{-0.9,-0.45}
$

2d)
%
$\begin{array}[t]{l}
 \Pr_{\Vec{AB}} \Vec{AC} = \Pr_{\VEC{2,1}} \VEC{1,2} = \frac{4}{5} \VEC{2,1} = \VEC{1.6,0.8} \\
 C' := A + \Pr_{\Vec{AB}} \Vec{AC} = (0,1) + \VEC{1.6,0.8} = (1.6, 1.8) \\
 \Vec{CC'} = C'-C = (1.6, 1.8) - (1,3) = \VEC{0.6, -1.2} \\
 C'' := C' + \Vec{C'C''} = C' + \Vec{CC'} = (1,3) + \VEC{0.6, -1.2} = (2.2,0.6) \\
 \end{array}
$

\newpage

3)
%
%L A = v(1,  1)
%L B = v(0, -1)
%L BB = v(2, 3)
%L C = Ellipse.newcircle(A, 16/15)
%L C = Ellipse.newcircle(A, 1)
%L l = Line.new(v(0,  -1), v(1, 3/4), -3, 4)
%L r = Line.new(v(0, 7/3), v(1, -4/3), -3, 4)
%L s = Line.new(BB, v(1, 3/4), -4, 3)
%L P = v(8/5, 1/5)
\pu
$\tikzp{[scale=0.75,auto]
    \mygrid (-1,-1) (4,3);
    \mydraw C:draw();
    \mydraw l:draw();
    \mydraw r:draw();
    \mydraw s:draw();
    \myldot A     0 A;
    \myldot B     0 B;
    \myldot BB    0 B';
    \myldot P     0 P;
  }
$

a) $\begin{array}[t]{l}
   r = \setofxyst{3x-4y-4=0} = \setofxyst{y=\frac{3}{4}x-1} \\
   d((1,1), r) = \frac{5/4}{\sqrt{1+\frac{9}{16}}} = \frac{5/4}{\sqrt{25/16}} = \frac{5/4}{5/4} = 1 \\
   C = \setofxyst{(x-1)^2+(y-1)^2=1} \\
   \end{array}
   $

b) $\begin{array}[t]{l}
   B' := A + 2\Vec{BA} = (0,-1) + 2\VEC{1,2} = (2,3) \\
   s = \setofet{(2,3)+t\VEC{1,\frac{3}{4}}} = \setofxyst{y=\frac{3}{4}x+1.5} \\
   \end{array}
   $

\bsk
\bsk

4)
%L r = Line.new(v(-1, 2), v(2,-1), -3, 3)
%L s = Line.new(v(1,  3), v(1, -2), -3, 3)
\pu
$\tikzp{[scale=0.75,auto]
    \mygrid (-1,-1) (4,3);
    \mydraw r:draw();
    \mydraw s:draw();
    \myldot r:t(0) 270 t{=}0;
    \myldot r:t(1) 270 t{=}1;
    \myldot s:t(0)   0 k{=}0;
    \myldot s:t(1)   0 k{=}1;
  }
$

$r = \setofet{(-1,2)+t\VEC{2,-1}} = \setofxyst{y = 1.5 - \frac{x}{2}}$

$s = \setofek{(1,3)+k\VEC{1,-2}}  = \setofxyst{y = 5 - 2x}$

$P = (x,y) ∈ r∩s$

$1.5 - \frac{x}{2} = 5 - 2x \quad⇒\quad 1.5x = 3.5 \quad⇒\quad x=\frac{7}{3}$

$y = 5 - 2\frac{7}{3} = \frac{15}{3} - \frac{14}{3} = \frac{1}{3}
  \quad⇒\quad P=(\frac{7}{3},\frac{1}{3})
$

Sejam $\uu := \VEC{2,-1}$ e $\vv := \VEC{1,-2}$. Temos $\nm{\uu} = \nm{\vv}$, então

$b = \setofet{P+t(\uu+\vv)} = \setofet{(\frac{7}{3},\frac{1}{3})+t(3,-3)}$ e

$b' = \setofet{P+t(\uu-\vv)} = \setofet{(\frac{7}{3},\frac{1}{3})+t(1,1)}$

são bissetrizes de $r$ e $s$.




\newpage

5)
%
%L C1 = Ellipse.newcircle(v(0,3), 2)
%L C2 = Ellipse.newcircle(v(1,0), 1)
%L C3 = Ellipse.newcircle(v(1,0), 1)
\pu
$\tikzp{[scale=0.75,auto]
    \mygrid (-2,-1) (4,5);
    \mydraw C1:draw();
    \mydraw C2:draw();
    \mydraw C3:draw();
    \myldot C1.C0  270 C_{1\,0};
    \myldot C2.C0  90  C_{2\,0};
    \myldot C3.C0  270 C_{3\,0};
  }
$

$\begin{array}{lll}
C_1 &=& \setofeth{(0,3)+2\VEC{\cosθ,\senθ}} \\
    &=& \setofxyst{x^2+(y-3)^2=2^2} \\
C_2 &=& \setofxyst{x^2-2x+y^2=0} \\
    &=& \setofxyst{(x-1)^2-1+y^2=0} \\
    &=& \setofxyst{(x-1)^2+y^2=1} \\
C_3 &=& \setofxyst{(x-1)^2+y^2=1} \\
 \end{array}
$
\bsk
\bsk

6)
%
%L A, B, C, D = v(3,3), v(5,3), v(5,4), v(-1,1)
%L r = Line.new(v(0, 5/3), v(1, 2/3), -2, 7)
%L vv, ww = v(3, 2), v(-2, 3)
%L D1, D2, D3 = D+vv, D+2*vv, D+2*vv+ww
\pu
$\tikzp{[scale=0.5,auto]
    % \mygrid (-1,-1) (4,4);
    \mygrid (-2,0) (7,8);
    % \draw [mycurve]  \e{A} -- \e{B} -- \e{C} -- \e{D} -- \e{A};
    \mydraw r:draw();
    \draw [mycurve] \e{B} -- \e{A} -- \e{C} -- \e{B};
    \myldot A   270 A;
    \myldot B   270 B;
    \myldot C     0 C;
    \myldot D   270 D;
    \myldot D2    0 E;
    \myldot D3    0 F;
    \draw [mycurve] \e{D} -- \e{D2} -- \e{D3} -- \e{D};
  }
$

$r = \setofxyst{2x-3y+5=0} = \setofet{D+t\VEC{3,2}}$

$E := D + 2\VEC{3,2}$

$F := D + 2\VEC{3,2} + \VEC{-2,3}$

\end{document}



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