Analytical Plane Geometry

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 \( y=-3x-1 \).
\( \frac{3\sqrt{10}}5 \)
\( \frac{2\sqrt{10}}5 \)
\( \frac{4\sqrt{10}}5 \)
\( \frac{\sqrt{10}}5 \)

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 \( S \).
\( C=[8,5]\text{, } D=[4,8] \)
\( C=[7,5]\text{, } D=[4,8] \)
\( C=[8,5]\text{, } D=[4,7] \)
\( C=[4,8]\text{, } D=[8,5] \)