Analytical Plane Geometry

1103061207

Level: 
A
We are given the straight line \( m= \left\{[3-t,t]\text{, } t\in\mathbb{R} \right\} \) which intersects lines \( a \), \( b \), \( c \) in points \( A \), \( B \), \( C \) consecutively (see the picture). Find the values of a parameter \( t \) corresponding to these line intersections.
\( t_A=1, t_B=\frac32,\ t_C=2 \)
\( t_A=-1, t_B=-2,\ t_C=-3 \)
\( t_A=2, t_B=\frac32,\ t_C=1 \)
\( t_A=2, t_B=\frac52,\ t_C=3 \)

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.
\( q\colon y=-6x-10 \)
\( q\colon y=-5x-10 \)
\( q\colon y=-5x-8 \)
\( q\colon y=-6x-8 \)

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.

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, \( p=q \)

2010014209

Level: 
A
In the following list identify a vector having the same direction as the line passing through the points \(A\) and \(B\). \[ A = \left [4,1\right ],\ \qquad B = \left [3,2\right ] \]
\(\left (-1,1\right )\)
\(\left (1,1\right )\)
\(\left (7,3\right )\)
\(\left (5,5\right )\)