Analytic geometry in a plane

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 \( 2 \).
\( X_1 = \left[2;\frac12\right]\text{, }X_2 = \left[6;\frac52\right] \)
\( X_1 = \left[2;1\right]\text{, }X_2 = \left[6;5\right] \)
\( X_1 = \left[2;\frac14\right]\text{, }X_2 = \left[6;\frac54\right] \)
\( X_1 = \left[2;\frac32\right]\text{, }X_2 = \left[6;\frac72\right] \)

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 \( 1 \).
\( X_1 = \left[5;2\right]\text{, }X_2 = \left[9;4\right] \)
\( X_1 = \left[4;2\right]\text{, }X_2 = \left[8;4\right] \)
\( X_1 = \left[2;4\right]\text{, }X_2 = \left[6;4\right] \)
\( X_1 = \left[2;5\right]\text{, }X_2 = \left[4;9\right] \)

2010014204

Level: 
B
Find the distance between parallel lines \( p \) and \( q \) given by their parametric equations. \begin{align*} p\colon x&=3-2t, & q\colon x&=2+2s, \\ y&=-1+t;\ t\in\mathbb{R}; & y&=1-s;\ s\in\mathbb{R}. \end{align*}
\(\frac{3\sqrt{5}}5\)
\(-\frac{3\sqrt{5}}5\)
\(\sqrt{5}\)
\(\frac{\sqrt{5}}3\)

2010014206

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

2010014607

Level: 
B
Given points \(A = [3;3]\), \(B = [-5;3]\) and \(C = [-1;-1]\), find the length of the altitude of the triangle \(ABC\) through the point \(C\). Hint: The altitude through the point \(C\) of a triangle \(ABC\) is the perpendicular line segment drawn from the vertex \(C\) to the line containing the side \(AB\).
\(4\)
\(\frac43\)
\(6\)
\(\frac23\)

2010014608

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 \( A=[4;-1] \), and \( B=\left[-3;\frac32\right] \) (see the picture).
\( 14x-5y-43=0 \)
\( 5x-14y-52=0 \)
\( 14x+5y-13=0 \)
\( 5x+14+32=0 \)