Analytical Plane Geometry

9000149409

Level: 
B
Find all lines which are parallel to \(p\colon x - 3y + 2 = 0\) and the distance from every of these lines to \(p\) is \(\sqrt{10}\).
\(p_{1}\colon x - 3y + 12 = 0\), \(p_{2}\colon x - 3y - 8 = 0\)
\(p\colon x - 3y = 0\)
\(p\colon x - 3y + \sqrt{10} = 0\)
\(p_{1}\colon x - 3y + \sqrt{10} = 0\), \(p_{2}\colon x - 3y -\sqrt{10} = 0\)

9000149406

Level: 
B
Given points \(A = [2;-5]\), \(B = [2;3]\) and \(C = [-4;-1]\), 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 \(AB\).
\(6\)
\(\sqrt{2}\)
\(\frac{3} {2}\)
The points \(A\), \(B\), \(C\) do not define a triangle.

9000107505

Level: 
B
Find \(\cos \varphi \) where \(\varphi \) is the angle between the lines \(p\) and \(q\). \[ \begin{aligned}[t] p\colon x& = 1 + 4t, & \\y & = 3 - 3t;\ t\in \mathbb{R}; \\ \end{aligned} \quad q\colon x + y - 3 = 0 \]
\(\frac{7\sqrt{2}} {10} \)
\(- \frac{7} {5\sqrt{2}}\)
\(\frac{\sqrt{2}} {5} \)
\(\frac{\sqrt{2}} {10} \)