Analytical Plane Geometry

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

1103061301

Level: 
B
Let \( ABC \) be a triangle (see the picture). Find the standard form equations of the lines \( t \), \( v \) and \( o \), where \( t \) contains the median to \( AB \), \( v \) contains the altitude to \( AB \) and \( o \) is the line of symmetry of \( AB \). Choose the option with all equations correct.
\( t\colon 2x+y-10=0 ,\ v\colon 4x+y-16=0,\ o\colon 4x+y-20=0 \)
\( t\colon 2x+y-10=0,\ v\colon x-4y+13=0,\ o\colon x-4y-5=0 \)
\( t\colon x-2y-5=0,\ v\colon 4x+y-16=0,\ o\colon 4x+y-20=0 \)
\( t\colon x-2y-5=0,\ v\colon x-4y+13=0,\ o\colon x-4y-5=0 \)