2010014210
Level:
A
In the following list find the vector having the same direction as the line
\(p\).
\[
p\colon 2x +3y + 1 = 0
\]
\(\left (-3;2\right )\)
\(\left (2;3\right )\)
\(\left (2;-3\right )\)
\(\left (3;1\right )\)
Analytical plane geometry
2010014209
Level:
A
In the following list identify a vector having the same direction as the line passing through
the points \(A\)
and \(B\).
\[
A = \left [4;1\right ],\ \qquad B = \left [3;2\right ]
\]
\(\left (-1;1\right )\)
\(\left (1;1\right )\)
\(\left (7;3\right )\)
\(\left (5;5\right )\)
2010014208
Level:
C
Given lines \(p\)
and \(q\), find
\(m\in \mathbb{R}\) such that
the lines \(p\)
and \(q\)
are parallel.
\[
p\colon x +3y + 4= 0,\qquad q\colon mx -2y - 7= 0
\]
\( m=-\frac23 \)
\( m=6 \)
\( m=-\frac13 \)
\( m=\frac13 \)
2010014207
Level:
C
Given two points \(A = [2;1]\)
and \(B = [4;-2]\), identify a
number \(m\in \mathbb{R}\) such
that the point \(C = [1;m]\)
is on the line \(AB\).
\( m=\frac52 \)
\( m=-\frac12 \)
\( m=2 \)
\( m=\frac13 \)
2010014206
Level:
B
Let \( p \) be the line with the equation \( x+2y-1=0 \). Find the general form equations of all lines parallel to \( p \) such that their distance from \( p \) equals to \( \sqrt5 \).
\( x+2y-6=0;\ x+2y+4=0 \)
\( x+2y-1=0;\ x+2y+1=0 \)
\( 2x-y-6=0;\ 2x-y+4=0 \)
\( 2x-y-1=0;\ 2x-y+1=0 \)
2010014205
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.
\( p\colon 2x+5y-4=0 \)
\( p\colon 5x-2y=0 \)
\( p\colon 5x-2y-10=0 \)
\( p\colon 2x+5y+33=0 \)
2010014204
Level:
B
Find the distance between parallel lines \( p \) and \( q \) given by their parametric equations.
\begin{align*}
p\colon x&=3-2t, & q\colon x&=2+2s, \\
y&=-1+t;\ t\in\mathbb{R}; & y&=1-s;\ s\in\mathbb{R}.
\end{align*}
\(\frac{3\sqrt{5}}5\)
\(-\frac{3\sqrt{5}}5\)
\(\sqrt{5}\)
\(\frac{\sqrt{5}}3\)
2010014203
Level:
B
Find the distance between parallel lines \(p\) and \(q\), if they are given by their general form equations, where \(p\) is \(−2x−4y+8=0\) and \(q\) is \(−x−2y+3=0\).
\(\frac{\sqrt{5}}5\)
\(-\frac1{\sqrt{5}}\)
\(\frac{2\sqrt{5}}5\)
\(\frac13\)
2010014202
Level:
A
Determine the relative position of the lines \( p\colon 6x+4y+8=0 \) and \( q\colon y=-\frac32 x+2 \).
parallel different lines, \( p\parallel q;\ p\neq q \)
intersecting lines, \( p\cap q=\left\{\left[0;-2\right]\right\} \)
intersecting lines, \( p\cap q=\left\{\left[0;2\right]\right\} \)
identical lines, \( p=q \)
2010014201
Level:
A
Find the value of a parameter \( a \), such that the line \( ax-4y-12=0 \) is parallel to the line \( y=-\frac32 x+4 \).
\( a=-6\)
\( a=-\frac32\)
\( a=4\)
\( a=\frac23\)