Plane geometry

1103109006

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 \( \sqrt5 \).
\( x-2y+4=0;\ x-2y-6=0 \)
\( x-2y+\sqrt5=0;\ x-2y-\sqrt5=0 \)
\( x-2y-1+\sqrt5=0;\ x-2y-1-\sqrt5=0 \)
\( x-2y+6=0;\ x-2y-4=0 \)

1103109005

Level: 
B
Let \( p \) be the line with the equation \( x-2y+5=0 \) and let \( \vec{v} \) be the vector \( (3;-2) \) (see the picture). Find the general form equation of the line \( p' \) which is the image of the line \( p \) translated by the vector \( \vec{v} \).
\( p'\colon x-2y-2=0 \)
\( p'\colon 2x-4y-3=0 \)
\( p'\colon x-2y-1=0 \)
\( p'\colon 2x-4y+3=0 \)

1103109004

Level: 
B
Let \( p \) be the line with the equation \( x-2y-1=0 \) and let \( S \) be the point with coordinates \( [2;2] \) (see the picture). Find the general form equation of the line \( p' \) which is the image of the line \( p \) in the point symmetry with the centre in \( S \).
\( p'\colon x-2y+5=0 \)
\( p'\colon 2x-4y+9=0 \)
\( p'\colon x-2y+4=0 \)
\( p'\colon x-2y+6=0 \)

1103109003

Level: 
B
Let \( 2x+6y-5=0 \) be the line \( p \) and \( x+3y-4=0 \) be the line \( o \), where \( p \) and \( o \) are parallel (see the picture). Find the general form equation of a line \( p' \) which is the reflection of the line \( p \) about the line of symmetry \( o \).
\( p'\colon 2x+6y-11=0 \)
\( p'\colon 2x+6y-2=0 \)
\( p'\colon 2x+6y+5=0 \)
\( p'\colon -2x-6y-11=0 \)

1103109002

Level: 
B
Let \( A=[0;1] \), \( B=[4;-2] \) and \( S=[4;3] \) be the points (see the picture). Find the coordinates of the points \( C \) and \( D \) so that \( ABCD \) is a parallelogram with the centre \( S \).
\( C=[8;5]\text{, } D=[4;8] \)
\( C=[7;5]\text{, } D=[4;8] \)
\( C=[8;5]\text{, } D=[4;7] \)
\( C=[4;8]\text{, } D=[8;5] \)

1103109001

Level: 
B
Let \( A \) be the point \( [4;3] \) and let the line \( p \) has the equation \( x-y+3=0 \). Find the coordinates of the point \( A' \) which is a mirror reflection of \( A \) about the line of symmetry \( p \) (see the picture).
\( A'=[0;7] \)
\( A'=[1;8] \)
\( A'=[-1;8] \)
\( A'=[-1;7] \)

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