1003061206
Level:
A
Find the value of a parameter \( a \), such that the line \( ax-6y-15=0 \) is parallel to the line \( y=-\frac23 x+4 \).
\( a=-4 \)
\( a=4 \)
\( a=-\frac23 \)
\( a=-2 \)
Analytical plane geometry
1103061205
Level:
A
From the following list choose the equation of a straight line that passes through the given point \( K \) and is not perpendicular to the given line \( m \) (see the picture).
\( r\colon y=\frac23x-\frac{13}3 \)
\( p\colon 3x+2y-13=0 \)
\( s\colon y=-\frac32x+\frac{13}2 \)
$\begin{aligned}
q\colon x&=5+2t, \\
y&=-1-3t;\ t\in\mathbb{R}
\end{aligned}$
1103061204
Level:
A
From the following list choose the equation of a straight line that passes through the given point \( K \) and is not parallel to the given line \( m \) (see the picture).
\( g\colon y=-\frac32x+\frac{13}2 \)
\( b\colon 2x-3y-13=0 \)
\( f\colon y=\frac23x-\frac{13}3 \)
$\begin{aligned}
q\colon x&=5+3t, \\
y&=-1+2t;\ t\in\mathbb{R}
\end{aligned}$
1103061203
Level:
A
A straight line \( p \) is given by the point \( A \) and the direction angle \( \varphi \) (see the picture). Choose the equation of the line \( p \) in the slope-intercept form.
\( p\colon y=-\sqrt3x+3 \)
\( p\colon y=\sqrt3x+3 \)
\( p\colon y=1.7x+3 \)
\( p\colon y=-1.7x+3 \)
1103061202
Level:
A
A straight line \( p \) is given by the point \( A \) and the normal vector \( \vec{n} \) (see the picture). Find its equation in general form.
\( p\colon 2x-5y-6=0 \)
\( p\colon 2x+5y-6=0 \)
\( p\colon 5x-2y-15=0 \)
\( p\colon 5x+2y-15=0 \)
1103061201
Level:
A
From the following list, choose parametric equations, that do not represent the straight line passing through the points \( A \) and \( B \) (see the picture).
$\begin{aligned}
p\colon x&=2+4t, \\
y&=6+2t;\ t\in\mathbb{R}
\end{aligned}$
$\begin{aligned}
p\colon x&=2+2t, \\
y&=1+t;\ t\in\mathbb{R}
\end{aligned}$
$\begin{aligned}
p\colon x&=6+4t, \\
y&=3+2t;\ t\in\mathbb{R}
\end{aligned}$
$\begin{aligned}
p\colon x&=2-2t, \\
y&=1-t;\ t\in\mathbb{R}
\end{aligned}$
$\begin{aligned}
p\colon x&=4+4t, \\
y&=2+2t;\ t\in\mathbb{R}
\end{aligned}$
1103090806
Level:
A
We are given the line segment \( AB \):
\begin{align*}
x&=2+2t, \\
y&=-1+t;\ t\in [0;1],
\end{align*}
and the points \( K=\left[\frac72;-\frac14\right] \), \( L=[-2;-3] \) and \( M=\left[5;\frac12\right] \). Choose a picture where the mutual position of the points \( A \), \( B \), \( K \), \( L \), and \( M \) is indicated correctly.
1103090805
Level:
B
Find the straight lines passing through the coordinate origin at the distance of \( 2 \) from the point \( M=[0;4] \). Express their equations in the slope-intercept form.
\( y=\pm\sqrt3 x \)
\( y=\pm4 x \)
\( y=\pm\frac32 x \)
\( y=\pm2\sqrt3 x \)
1003090804
Level:
B
Find the distance between parallel lines \( p \) and \( q \) given by their parametric equations.
\begin{align*}
p\colon x&=3+3t, & q\colon x&=2-3s, \\
y&=-1+t;\ t\in\mathbb{R}; & y&=1-s;\ s\in\mathbb{R}.
\end{align*}
\( \frac{7\sqrt{10}}{10} \)
\( \frac{\sqrt{10}}{2} \)
\( \frac{\sqrt{10}}{5} \)
\( \frac{5\sqrt{10}}{2} \)
1003090803
Level:
B
Find the distance between parallel lines \( p \) and \( q \), if they are given by slope-intercept form equations, where \( p \) is \( y=-3x+5 \) and \( q \) is \( y=-3x-1 \).
\( \frac{3\sqrt{10}}5 \)
\( \frac{2\sqrt{10}}5 \)
\( \frac{4\sqrt{10}}5 \)
\( \frac{\sqrt{10}}5 \)