Analytical plane geometry

1003090802

Level: 
B
Find the distance between parallel lines \( p \) and \( q \), if they are given by their general form equations, where \( p \) is \( 2x-4y+5=0 \) and \( q \) is \( x-2y+3=0 \).
\( \frac{\sqrt5}{10} \)
\( \frac{11\sqrt5}{10} \)
\( \frac{3}{2\sqrt5} \)
\( \frac{3\sqrt5}{10} \)

1103090801

Level: 
B
Find a general form equation of the straight line that passes through the point \( M=[2;3] \) and is parallel with the line of symmetry of the line segment \( AB \), where \( A=[-1;4] \), and \( B=\left[\frac52;-3\right] \) (see the picture).
\( x-2y+4=0 \)
\( 2x+y-7=0 \)
\( 3x+2y-12=0 \)
\( 2x-3y+5=0 \)

1103109008

Level: 
B
Let \( p \) be the line with the equation \( x-2y-1=0 \). Find the coordinates of all points lying on the line \( p \) such that their distance from the line \( y=3 \) equals to \( 1 \).
\( X_1 = \left[5;2\right]\text{, }X_2 = \left[9;4\right] \)
\( X_1 = \left[4;2\right]\text{, }X_2 = \left[8;4\right] \)
\( X_1 = \left[2;4\right]\text{, }X_2 = \left[6;4\right] \)
\( X_1 = \left[2;5\right]\text{, }X_2 = \left[4;9\right] \)

1103109007

Level: 
B
Let \( p \) be the line with the equation \( x-2y-1=0 \). Find the coordinates of all points lying on the line \( p \) such that their distance from the line \( x=4 \) equals to \( 2 \).
\( X_1 = \left[2;\frac12\right]\text{, }X_2 = \left[6;\frac52\right] \)
\( X_1 = \left[2;1\right]\text{, }X_2 = \left[6;5\right] \)
\( X_1 = \left[2;\frac14\right]\text{, }X_2 = \left[6;\frac54\right] \)
\( X_1 = \left[2;\frac32\right]\text{, }X_2 = \left[6;\frac72\right] \)

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).
\( A'=[0;7] \)
\( A'=[1;8] \)
\( A'=[-1;8] \)
\( A'=[-1;7] \)