Plane geometry

1103109104

Level: 
C
Let \( 2x-3y+6=0 \) be the equation of the line \( p \) and let \( M \) be the point \( [5;3] \). Find equations of all lines passing through \( M \) and intersecting \( p \) at an angle of \( 45^{\circ} \) (see the picture).
\( x+5y-20=0;\ 5x-y-22=0 \)
\( x+6y-23=0;\ 6x-y-27=0 \)
\( x+4y-17=0;\ 4x-y-16=0 \)
\( x+5y-28=0;\ 5x-y-10=0 \)

1103109103

Level: 
C
Let \( y=-\frac{\sqrt3}3x+1 \) be the equation of the line \( p \) and let \( M \) be the point \( [0;-3] \). Find equations of all lines passing through \( M \) and intersecting \( p \) at an angle of \( 60^{\circ} \) (see the picture).
\( x=0;\ y=\frac{\sqrt3}3x-3 \)
\( y=0;\ y=\frac{\sqrt3}3x-3 \)
\( y=0;\ y=x-3 \)
\( x=0;\ y=\sqrt3x-3 \)

1103109102

Level: 
C
Let \( p \) and \( q \) be intersecting lines with the equations \( y=\frac{\sqrt3}3x \) and \( x=0 \) respectively. Find equations of lines \( o_1 \) and \( o_2 \) that are lines of symmetry of the angles contained between \( p \) and \( q \) (see the picture).
\( y=\sqrt3x;\ y=-\frac{\sqrt3}3x \)
\( y=2x;\ y=-\frac12x \)
\( y=\sqrt2x;\ y=-\frac{\sqrt2}2x \)
\( y=3x;\ y=-\frac13x \)

1103109101

Level: 
C
Find equations of all lines at the distance \( \sqrt{10} \) from the point \( M=[5;4] \) which are perpendicular to the line \( p \) with the equation \( 2x+6y-3=0 \) (see the picture).
\( 3x-y-1=0;\ 3x-y-21=0 \)
\( 3x-y+1=0;\ 3x-y-18=0 \)
\( x+3y+1=0;\ x+3y+21=0 \)
\( x+3y-1=0;\ x+3y-18=0 \)

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.
\( t_A=1; t_B=\frac32;\ t_C=2 \)
\( t_A=-1; t_B=-2;\ t_C=-3 \)
\( t_A=2; t_B=\frac32;\ t_C=1 \)
\( t_A=2; t_B=\frac52;\ t_C=3 \)

1103061205

Level: 
A
From the following list choose the equation of a straight line that passes through the given point \( K \) and is not perpendicular to the given line \( m \) (see the picture).
\( r\colon y=\frac23x-\frac{13}3 \)
\( p\colon 3x+2y-13=0 \)
\( s\colon y=-\frac32x+\frac{13}2 \)
$\begin{aligned} q\colon x&=5+2t, \\ y&=-1-3t;\ t\in\mathbb{R} \end{aligned}$

1103061204

Level: 
A
From the following list choose the equation of a straight line that passes through the given point \( K \) and is not parallel to the given line \( m \) (see the picture).
\( g\colon y=-\frac32x+\frac{13}2 \)
\( b\colon 2x-3y-13=0 \)
\( f\colon y=\frac23x-\frac{13}3 \)
$\begin{aligned} q\colon x&=5+3t, \\ y&=-1+2t;\ t\in\mathbb{R} \end{aligned}$

1103061203

Level: 
A
A straight line \( p \) is given by the point \( A \) and the direction angle \( \varphi \) (see the picture). Choose the equation of the line \( p \) in the slope-intercept form.
\( p\colon y=-\sqrt3x+3 \)
\( p\colon y=\sqrt3x+3 \)
\( p\colon y=1.7x+3 \)
\( p\colon y=-1.7x+3 \)