Analytic geometry in a space

1003188904

Level: 
A
Determine the relative position of the plane \( \rho \) with general equation \( 7x-2y+z-2=0 \) and the straight line \( p \) with parametric equations: \[ \begin{aligned} x&=3+t, \\ y&=-5-2t, \\ z&=3-11t;\ t\in\mathbb{R}. \end{aligned} \]
\( p\parallel \rho\text{, }p\not{\!\!\subset}\rho \)
\( p \subset \rho \)
\( p \) is intersecting the plane \( \rho \)

1003188905

Level: 
A
Determine the relative position of the plane \( \rho \) with general equation \( 5x-4y+z-4=0 \) and the straight line \( p \) with parametric equations: \[ \begin{aligned} x&=-1+t,\\ y&=2-2t,\\ z&=3+t;\ t\in\mathbb{R}. \end{aligned} \]
\( p \) is intersecting \( \rho \)
\( p\parallel \rho\text{, } p\not{\!\!\subset}\rho \)
\( p \subset \rho \)

1003188906

Level: 
A
Let there be planes \( \alpha \), \( \beta \), \( \gamma \) and \( \delta \) defined by their general equations: \[ \begin{aligned} &\alpha\colon \frac23x-4y+6z-\frac83=0; \\ &\beta\colon x-2y+3z-4=0; \\ &\gamma\colon 2x-12y+18z-4 =0; \\ &\delta\colon x-6y+9z-4 =0. \end{aligned} \] Out of the following statements, select the one that is not true.
\( \alpha \parallel\delta\text{, }\alpha\neq\delta \)
Planes \( \beta \) and \( \delta \) are intersecting.
\( \gamma\parallel\delta\text{, }\gamma\neq\delta \)
Planes \( \alpha \) and \( \beta \) are intersecting.
\( \alpha = \delta \)

1003188907

Level: 
A
We are given two intersecting planes \( x-6y+9z-4=0 \) and \( x-2y+3z-4=0 \). Find the parametric equations of their line of intersection \( p \).
\( \begin{aligned} p\colon x&=4, \\ y&=\phantom{4+}\ 3t, \\ z&=\phantom{4+}\ 2t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p\colon x&=4+t , \\ y&=\phantom{4+}\ 3t , \\ z&=\phantom{4+}\ 2t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p\colon x&=4, \\ y&=\frac32+3t, \\ z&=1+2t;\ t\in\mathbb{R} \end{aligned} \)
\( \begin{aligned} p\colon x&=4+t, \\ y&=\frac32+3t, \\ z&=1+2t;\ t\in\mathbb{R} \end{aligned} \)

1103188705

Level: 
A
Find parametric equations of the line \( p \) that passes through the point \( K=[4;2;3] \), is parallel to the \( xy \)-coordinate plane, and is intersecting the \( z \)-axis.
$\begin{aligned} p\colon x&=4+2t, \\ y&=2+t, \\ z&=3;\ t\in\mathbb{R} \end{aligned}$
$\begin{aligned} p\colon x&=4+2t, \\ y&=2+t, \\ z&=3+t;\ t\in\mathbb{R} \end{aligned}$
$\begin{aligned} p\colon x&=4, \\ y&=2, \\ z&=3+3t;\ t\in\mathbb{R} \end{aligned}$
$\begin{aligned} p\colon x&=4-2t, \\ y&=2-4t, \\ z&=3t;\ t\in\mathbb{R} \end{aligned}$

1103188706

Level: 
A
We are given points \( A=[2;4;0] \) and \( B=[4;7;6] \). Find parametric equations of a line \( q \), which is the orthogonal projection of the line \( AB \) into the coordinate plane \( xy \).
$\begin{aligned} p\colon x&=4+2t, \\ y&=7+3t, \\ z&=0;\ t\in\mathbb{R} \end{aligned}$
$\begin{aligned} p\colon x&=2+4t, \\ y&=4+7t, \\ z&=6t;\ t\in\mathbb{R} \end{aligned}$
$\begin{aligned} p\colon x&=4+2t, \\ y&=7+3t, \\ z&=6;\ t\in\mathbb{R} \end{aligned}$
$\begin{aligned} p\colon x&=2-2t, \\ y&=4-3t, \\ z&=-6t;\ t\in\mathbb{R} \end{aligned}$