Warning: this is an htmlized version!
The original is across this link,
and the conversion rules are here.
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% (defun e () (interactive) (find-LATEX "2016-2-GA-VS.tex"))
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%   file:///home/edrx/LATEX/2016-2-GA-VS.pdf
%               file:///tmp/2016-2-GA-VS.pdf
%           file:///tmp/pen/2016-2-GA-VS.pdf
% http://angg.twu.net/LATEX/2016-2-GA-VS.pdf
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\def\V(#1){\VEC{#1}}

\setlength{\parindent}{0em}
\def\T(Total: #1 pts){{\bf(Total: #1 pts)}}
\def\T(Total: #1 pts){{\bf(Total: #1)}}
\def\B       (#1 pts){{\bf(#1 pts)}}
% Usage:
% 1) \T(Total: 2.34 pts) Foo
% a) \B(0.45 pts) Bar



%   ____      _                    _ _           
%  / ___|__ _| |__   ___  ___ __ _| | |__   ___  
% | |   / _` | '_ \ / _ \/ __/ _` | | '_ \ / _ \ 
% | |__| (_| | |_) |  __/ (_| (_| | | | | | (_) |
%  \____\__,_|_.__/ \___|\___\__,_|_|_| |_|\___/ 
%                                                

{\setlength{\parindent}{0em}
\footnotesize
\par Geometria Analítica
\par PURO-UFF - 2016.2
\par VS - 25/jan/2017 - Eduardo Ochs
\par Respostas sem justificativas não serão aceitas.
\par Proibido usar quaisquer aparelhos eletrÃ∧nicos.
% \par Versão: 14/mar/2016
% \par Links importantes:
% \par \url{http://angg.twu.net/2015.2-C2.html} (página do curso)
% \par \url{http://angg.twu.net/2015.2-C2/2015.2-C2.pdf} (quadros)
% \par \url{http://angg.twu.net/LATEX/2015-2-C2-material.pdf}
% \par {\tt eduardoochs@gmail.com} (meu e-mail)
}

\bsk
\bsk

1) \T(Total: 3.0 pts) Sejam:
%
$$\begin{array}{rclcl}
    A &:=& (2,2),\\ 
    B &:=& (0,2),\\ 
  r_m &:=& \setofxyst{y=2+mx}, \quad (m∈\R)\\
  r_∞ &:=& \setofxyst{x=0}, \\
  C_m &\text{é}& \text{o ponto de $r_m$ mais próximo de $A$}.
  \end{array}
$$

a) \B(0.2 pts) Represente graficamente $A$, $B$, $r_0, r_1, r_{-1}, r_∞$.

b) \B(0.4 pts) Encontre e represente graficamente $C_0, C_1, C_{-1}, C_∞$.

c) \B(1.0 pts) Calcule e represente graficamente $C_{1/3}$.

d) \B(1.0 pts) Encontre a equação de uma cÃ∧nica que contenha $C_0,
C_1, C_{-1}, C_∞, C_{1/3}$.

e) \B(0.4 pts) Verifique que $C_0, C_1, C_{-1}, C_∞, C_{1/3}$ obedecem
a equação da cÃ∧nica.




\bsk
\bsk

2) \T(Total: 2.0 pts) Seja $H = \setofxyst{(\frac y2 - \frac x4)(\frac
  y2 + \frac x4) = 1}$.

a) \B(0.2 pts) Represente graficamente as assíntotas de $H$.

b) \B(0.2 pts) Dê as equações das assíntotas de $H$.

c) \B(0.8 pts) Encontre dois pontos de $H$.

d) \B(0.8 pts) Represente graficamente $H$.


\bsk
\bsk


3) \T(Total: 1.5 pts) Sejam $A=(2,0,0)$, $B=(0,3,0)$, $C=(2,3,4)$ e
seja $Ï€$ o plano contendo $A$, $B$ e $C$.

a) \B(0.5 pts) Se $π'=\setofxyzst{ax+by+cz=d}$ e $π'=π$, quem são $a$, $b$, $c$, $d$?

b) \B(1.0 pts) Encontre dois pontos de $π∩π''$, onde $π''=\setofxyzst{x+y=4}$.

\bsk
\bsk




4) \T(Total: 3.5 pts) Um dos ``usos do `$×$'\,'' na folha 35 é o
seguinte: se
%
$$\begin{array}{rclcl}
  r  &:=& \setof{A+t\uu}{t∈\R}, \\
  r' &:=& \setof{B+t'\vv}{t'∈\R}, \\
  C_\aa &:=& A+\aa\uu, \\ 
  D_\bb &:=& B+\bb\vv \\ 
  \end{array}
$$
%
então para encontrarmos os pontos onde $r$ e $r'$ ficam mais próximas
basta resolver $\Vec{C_\aa D_\bb} × (\uu×\vv) = \V(0,0,0)$; aí a gente
encontra $\aa$ e $\bb$, e os pontos são $C_\aa∈r$ e $D_\bb∈r'$.

a) \B(1.0 pts) Use isto para encontrar $C_\aa$ e $D_\bb$ no caso em
que $A=(1,1,0)$, $\uu=\V(2,0,0)$, $\vv=\V(2,3,0)$, $B=A+\V(2,3,4)$.

b) \B(2.5 pts) Use isto para encontrar $C_\aa$ e $D_\bb$ no caso em
que $r$ passa por $(1,0,0)$ e $(0,1,0)$ e $r'$ passa por $(1,1,0)$ e $(0,1,1)$.

{\sl Confira os seus resultados!!!}




\end{document}

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