Analytical plane geometry

1003061306

Level: 
A
Determine the relative position of the lines \( p\colon 2x-3y+7=0 \) and \[ \begin{aligned} q\colon x& =2+t, \\ y& = -3t, \end{aligned} \] where \( t\in\mathbb{R} \).
intersecting lines, \( p\cap q=\left\{\left[1;3\right]\right\} \)
identical lines, \( p=q \)
parallel different lines, \( p\parallel q;\ p\neq q \)
intersecting lines, \( p\cap q=\left\{\left[7;7\right]\right\} \)

1003061305

Level: 
A
Determine the relative position of the lines \( p\colon 4x+6y-5=0 \) and \( q\colon y=-\frac23 x-6 \).
parallel different lines, \( p\parallel q;\ p\neq q \)
identical lines, \( p=q \)
intersecting lines, \( p\cap q=\left\{\left[0;\frac54\right]\right\} \)
intersecting lines, \( p\cap q=\left\{\left[0;\frac56\right]\right\} \)

1003061304

Level: 
A
Determine the relative position of the lines \( p\colon4x-3y+9=0 \) and \[ \begin{aligned} q\colon x&=6+3t, \\ y&=11+4t, \end{aligned} \] where \( t\in\mathbb{R}\).
identical lines, \( p=q \)
parallel different lines, \( p\parallel q;\ p\neq q \)
intersecting lines, \( p\cap q=\{[0;3]\} \)
intersecting lines, \( p\cap q=\{[6;11]\} \)

1103061303

Level: 
A
Let there be a straight line \( p\colon 5x-y-10=0 \). Choose the equation of a straight line \( q \) that passes through the point \( A=[-2;2] \) and intersects with \( p \) on \( y \)-axis.
\( q\colon y=-6x-10 \)
\( q\colon y=-5x-10 \)
\( q\colon y=-5x-8 \)
\( q\colon y=-6x-8 \)

1103061301

Level: 
B
Let \( ABC \) be a triangle (see the picture). Find the standard form equations of the lines \( t \), \( v \) and \( o \), where \( t \) contains the median to \( AB \), \( v \) contains the altitude to \( AB \) and \( o \) is the line of symmetry of \( AB \). Choose the option with all equations correct.
\( t\colon 2x+y-10=0 ;\ v\colon 4x+y-16=0;\ o\colon 4x+y-20=0 \)
\( t\colon 2x+y-10=0;\ v\colon x-4y+13=0;\ o\colon x-4y-5=0 \)
\( t\colon x-2y-5=0;\ v\colon 4x+y-16=0;\ o\colon 4x+y-20=0 \)
\( t\colon x-2y-5=0;\ v\colon x-4y+13=0;\ o\colon x-4y-5=0 \)

9000151306

Level: 
B
Find the angle \(\varphi \) between the lines \(p\) and \(q\) given by their parametric equations. \[ p\colon \begin{aligned}[t] x& = 1 - t, & \\y& = 2 + t;\ t\in \mathbb{R}, \\ \end{aligned}\qquad q\colon \begin{aligned}[t] x& = 4 - k, & \\y& = 5 + k;\ k\in \mathbb{R}. \\ \end{aligned} \]
\(0^{\circ }\)
\(90^{\circ }\)
\(60^{\circ }\)
\(30^{\circ }\)