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%   ____      _                    _ _           
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{\setlength{\parindent}{0em}
\footnotesize
\par Geometria AnalÃtica
\par PURO-UFF - 2016.1
\par P2 - 28/jul/2016 - Eduardo Ochs
\par Respostas sem justificativas nÃo serÃo aceitas.
\par Proibido usar quaisquer aparelhos eletrÃ∧nicos.
\ssk
\par Links importantes:
\par \url{http://angg.twu.net/2016.1-GA.html} (pÃgina do curso)
\par \url{http://angg.twu.net/2016.1-GA/2016.1-GA.pdf} (quadros)
\par \url{http://angg.twu.net/LATEX/2016-1-GA-P2.pdf} (esta prova, com gabarito)
% \par \url{http://angg.twu.net/LATEX/2016-1-GA-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

1) \T(Total: 1.5 pts) Em cada um dos itens abaixo encontre 3 pontos,
$P_1$, $P_2$, $P_3$, da parÃbola $S=\setofst{Pâ\R^2}{d(P,F)=d(P,d)}$ e
a equaÃÃo de {\sl alguma} parÃbola que passa por estes pontos.

a) \B(0.2 pts) $F=(0,1)$, $d:y=-1$

b) \B(0.3 pts) $F=(0,2)$, $d:y=-2$

c) \B(1.0 pts) $F=(4,2)$, $d:x=0$

\bsk

2) \T(Total: 1.5 pts) Em cada um dos itens abaixo encontre 4 pontos
$P_1$, $P_2$, $P_3$, $P_4$ da elipse
$E=\setofst{Pâ\R^2}{d(P,d)=2d(P,F)}$ e a equaÃÃo de alguma elipse que
passa por estes pontos.

a) \B(0.5 pts) $F=(0.5,0)$, $d:x=2$

b) \B(1.0 pts) $F=(0,0)$, $d:y=3$



\bsk


% (find-es "ipython" "2016.1-GA-P2")

3) \T(Total: 1.0 pts) FaÃa um esboÃo da cÃ∧nica com equaÃÃo $(x-3)^2 -
(3y+3)^2 - 1 = 0$.



\bsk



% (find-es "ipython" "2016.1-GA-P2")

4) \T(Total: 4.0 pts) Sejam $r : (2+t, 1+2t, 11-4t)$ e $r' : (1+3u, 2-u, 6+9u)$.

a) \B(1.0 pts) Mostre que $r$ e $r'$ sÃo coplanares.

b) \B(1.0 pts) Encontre a equaÃÃo do plano $Ï$ contendo $r$ e $r'$.

c) \B(1.0 pts) Sejam $P=(4,0,4)$, $P'$ o ponto de $Ï$ mais prÃximo de
$P$, e $P''$ o ponto simÃtrico a $P$ com relaÃÃo a $Ï$. Dê as
coordenadas de $P'$ e $P''$.

d) \B(1.0 pts) Calcule $d(P,Ï)$.





\bsk



% (find-es "ipython" "2016.1-GA-P2")

5) \T(Total: 2.0 pts) Sejam $Ï : x+y+2z = 4$, $Ï' : z-4y=8$.

a) \B(1.0 pts) Encontre uma reta paralela a $Ï$ e $Ï'$ que passa por $P=(2,3,4)$.

b) \B(1.0 pts) Dê a equaÃÃo da reta $r=ÏâÏ'$.



\bsk
\bsk




Algumas fÃrmulas:

$[\uu,\vv,\ww] = \psm{u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ w_1 & w_2 & w_3 \\}
 \qquad
 \vsm{a & b & c \\ d & e & f \\ g & h & i \\} =
 \sm{aei + bfg + cdh \\ - afh - bdi - ceg}
 \qquad |[\uu,\vv,\ww]| = (\uuÃ\vv)Â\ww
$

$\uuÃ\vv = \vsm{u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \\ \ii & \jj & \kk \\}
         = {\scriptstyle \VEC{u_2v_3 - u_3v_2, u_3v_1 - u_1v_3, u_1v_2 - u_2v_1}}
$



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Mini-gabarito:

(incompleto e ainda nÃo revisado - contÃm erros!)

\bsk

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1a) $\bsm{P_{-1}=(-2,1) & & P_1=(2,1) \\ & P_0=(0,0) & \\}$; $S:y=x^2/2$

1b) $\bsm{P_{-1}=(-4,2) & & P_1=(4,2) \\ & P_0=(0,0) & \\}$; $S:y=x^2/8$

1c) $\bsm{ & P_{-1}=(4,6) \\ P_0=(2,2) & \\ & P_1=(4,-2) & \\}$; $S:(x-2)=(y-2)^2/8$

\bsk
\bsk

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2a) $\bsm{      & P_2=(0,â3/2) &      \\
          P_2=(-1,0) & & P_3=(1,0) \\
                & P_4=(0,-â3/2) &      \\
         }; \quad E:x^2+(\frac y {â3/2})^2 = 1$

2b) $\bsm{      & P_2=(-1,â3) &      \\
          P_2=(-3,0) & & P_3=(1,0) \\
                & P_4=(-1,â3) &      \\
         }; \quad E:(\frac{x+1}2)^2+(\frac y {â3})^2 = 1$

\bsk
\bsk

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3) $H: (x-3)^2 - (3y+3)^2 = 1$ Ã uma hipÃrbole.

$H_0: (x-3)^2 - (3y+3)^2 = 0$ sÃo as assÃntotas de $H$.

Como $(x-3)^2 - (3y+3)^2 = ((x-3) + (3y+3)) ((x-3) - (3y+3))$, sejam

$r:(x-3) + (3y+3) = 0$ e

$r':(x-3) - (3y+3) = 0$; temos

$H_0 = râªr'$,

$r: x+3y=0$ \;\;\;\;\;\;\;\;\;\; (ou: $r:y=-\frac x3$),

$r': x-3y-6=0$ \;\;\; (ou: $r':y=\frac x3 - 2$).

$r$ e $r'$ se intersectam em $x-3=0$ e $3y+3=0$, ou seja, em $(x,y)=(3,-1)$.

Pontos Ãbvios de $H$: $(x-3)^2=1$, $(3y+3)=0$; $x-3=\pm1$, $x=3\pm1$, $y=-1$.

$(2,-1)âH$, $(4,-1)âH$.


\newpage

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4a) $r = \setofexpron{A+t\vv}{t}$ e $r' = \setofexpron{B+u\vv'}{u}$, onde

$A = (2, 1, 11)$, $\vv = \VEC{1,2,-4}$,

$B = (1, 2, 6)$, $\vv' = \VEC{3, -1, 9}$.

$r$ e $r'$ sÃo coplanares se $\uu$, $\vv$ e $\vec{AB}$ sÃo coplanares.

$|[\uu, \vv, \vec{AB}]| = \vsm{1 & 2 & -4 \\ 3 & -1 & 9 \\ -1 & 1 & -5}
 = \sm{1Â(-1)Â(-5) + 2Â9Â(-1) + (-4)Â3Â1 \\ - (-4)Â(-1)Â(-1) - 2Â3Â(-5) - 1Â9Â1}
 = \sm{5 - 18 - 12 \\ + 4 + 30 - 9} = 0
$.

4b) Sejam
$\nn = \uuÃ\vv = \sm{\VEC{1,2,-4} \\ Ã \VEC{3,-1,9}} = \VEC{14, -21, -7}
     = 7\VEC{2,-3,-1}
    $

EntÃo $Ï=\setofxyzst{2x-3y-z=d}$,

e como $AâÏ$ temos $d=2Â2-3Â1-11=-10$,

e portanto $Ï=\setofxyzst{2x-3y-z=-10}$.

4c) Sejam $\nn' = \frac17\nn =\VEC{2,-3,-1}$, $s = \setofexpron{P+t\nn'}{t}$. EntÃo $P'âsâÏ$.

$P'=(4,0,4)+t\VEC{2,-3,-1} = (4+2t,-3t,4-t)$ obedece $2x-3y-z=-10$,

portanto $-10 = 2(4+2t)-3(-3t)-(4-t) = 8+4t +9t -4 + t = 14t+4$,

$14t=-14$, $t=-1$,

$P' = (4+2(-1),-3(-1),4-(-1)) = (2,3,5)$,

$\vec{PP'} = \VEC{-2,3,1}$,

$P'' = P'+\vec{PP'} = (2,3,5) + \VEC{-2,3,1} = (0,6,6)$.

4d) $d(P,Ï) = d(P,P') = ||\VEC{-2,3,1}|| = \sqrt{4+9+1} = \sqrt{14}$.




\bsk
\bsk

5a) Vetor normal a $Ï$: $\nn = \VEC{1,1,2}$.

Vetor normal a $Ï'$: $\nn' = \VEC{0,-4,1}$.

Vetor paralelo a $Ï$ e $Ï'$: $\vv = \nnÃ\nn' = \VEC{9, -1, -4}$.

Reta que queremos: $\setofexpron{(2,3,4)+t\VEC{9, -1, -4}}{t}$.

5b) Se $Q=(x,y,0)$ pertence a $r=ÏâÏ'$, entÃo $x+y=4$, $-4y=8$,

$y=-2$, $x=6$, $Q=(6,-2,0)$,

$r=\setofexpron{(6,-2,0)+t\VEC{9, -1, -4}}{t}$.

\end{document}

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