Warning: this is an htmlized version!
The original is across this link,
and the conversion rules are here.
% (find-angg "LATEX/2016-2-GA-P1.tex")
% (defun c () (interactive) (find-LATEXsh "lualatex -record 2016-2-GA-P1.tex"))
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% (find-sh0 "cp -v  ~/LATEX/2016-2-GA-P1.pdf /tmp/pen/")
%   file:///home/edrx/LATEX/2016-2-GA-P1.pdf
%               file:///tmp/2016-2-GA-P1.pdf
%           file:///tmp/pen/2016-2-GA-P1.pdf
% http://angg.twu.net/LATEX/2016-2-GA-P1.pdf
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%
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%
\begin{document}

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\def\V(#1){\VEC{#1}}





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

{\setlength{\parindent}{0em}
\footnotesize
\par Geometria Analítica
\par PURO-UFF - 2016.2
\par P1 - 16/nov/2016 - 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

\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



% (find-angg "LATEX/2015-2-GA-P2.tex")

1) \T(Total: 1.0 pts) Sejam
%
$$\begin{array}{rcl}
  r &=& \setofst {(1,2)+t\V(-1,2))} {t∈\R}, \\
  s &=& \setofst {(0,4)+u\V(2,-4))} {u∈\R}. \\
  \end{array}
$$

a) \B(0.2 pts) Represente $r$ e $s$ graficamente.

b) \B(0.8 pts) Escolha dois pontos diferentes de $r=s$ e dê as
coordenadas e os valores de $t$ e $u$ associados a cada um.


\bsk
\bsk

2) \T(Total: 2.0 pts) Sejam
%
$$\begin{array}{rcl}
    r &=& \setofst {(1,2)+t\V(3,4)} {t∈\R}, \\
  s_a &=& \setofxyst {y=5+ax}. \\
  \end{array}
$$

a) \B(1.0 pts) Encontre o valor de $a$ que faz com que $r$ e $s_a$ sejam
ortogonais.

b) \B(1.0 pts) Calcule $P∈r∩s_a$, onde o $a$ é o do item anterior.
Represente tudo graficamente.





\bsk
\bsk

3) \T(Total: 2.0 pts) Verdadeiro ou falso? Justifique.

a) \B(1.0 pts) $\Pr_{2\uu} (\ww) = 2(\Pr_{\uu} \ww)$

b) \B(1.0 pts) $\Pr_{\uu} (3\ww) = 3(\Pr_{\uu} \ww)$




\bsk
\bsk

4) \T(Total: 3.0 pts) Sejam
%
$$\begin{array}{rcl}
  r &=& \setofxyst{y=1+\frac34 x}, \\
  s &=& \setofexprt{(0,1)+t\VEC{2,1}}. \\
  \end{array}
$$

a) \B(0.3 pts) Calcule $d((0,3),r)$.

b) \B(1.2 pts) Encontre os dois pontos $P_1,P_2∈s$ que estão a distância 1 de $r$.

c) \B(1.5 pts) Encontre as duas retas, $r'$ e $r''$, que são paralelas
a $r$ e tais que $d(r,r') = d(r,r'') = 1$.


\bsk
\bsk

5) \T(Total: 2.0 pts) Sejam $C$ o círculo de com $C_0=(0,5)$ e $R=5$,
e $C'$ o círculo de com $C'_0=(1,0)$ e $R'=1$.

a) \B(0.2 pts) Obtenha as equações dos dois círculos.

b) \B(0.2 pts) Subtraia as duas equações para obter a equação de uma
reta $r$. Defina $r$ formalmente (como conjunto).

c) \B(1.0 pts) Encontre as coordenadas dos dois pontos $\{I,I'\}=C
\cap C' = C \cap r = C' \cap r$.

d) \B(0.6 pts) Verifique que os seus $I$ e $I'$ pertencem a $C$ e
$C'$.




\newpage

{\bf Mini-gabarito:}

{\footnotesize
(Complementa o que foi discutido em sala em 21/nov/2016:
\par \url{http://angg.twu.net/2016.2-GA/20161121_GA1.jpg}
\par \url{http://angg.twu.net/2016.2-GA/20161121_GA2.jpg})
}

\msk

2a) $a=-\frac34$

2b) $t = \frac{9}{25} = 0.36$, $(x,y) = (\frac{52}{25}, \frac{86}{25}) = (2.08, 3.44)$

\msk

4a) $d((0,3),r) = \frac 8 5$

4b) $P_1 = (-5,-\frac32)$, $P_1 = (5,\frac72)$

4c) $r': y = \frac34 x + \frac94$, $r'': y = \frac34 x - \frac14$

\msk

5c) $I=(0,0)$, $I'=(\frac{25}{13}, \frac{5}{13})$

\bsk
\bsk
\bsk


{\footnotesize
\par Links importantes:
\par \url{http://angg.twu.net/2016.2-GA.html} (página do curso)
\par \url{http://angg.twu.net/2016.2-GA/2016.2-GA.pdf} (quadros)
\par \url{http://angg.twu.net/LATEX/2016-2-GA-algebra.pdf} (material extra)
\par \url{http://angg.twu.net/LATEX/2016-2-GA-P1.pdf} (esta prova)
\par {\tt eduardoochs@gmail.com} (meu e-mail)
}


\end{document}

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