Analytical Plane Geometry

9000151302

Level: 
B
Find the angle \(\varphi \) between the lines \(p\) and \(q\) given by their parametric equations. \[ p\colon \begin{aligned}[t] x& = 1 + 2t, & \\y& = 3 - 3t,\ t\in \mathbb{R}, \\ \end{aligned}\qquad q\colon \begin{aligned}[t] x& = 2 - k, & \\y& = 3 + k,\ k\in \mathbb{R} \\ \end{aligned} \]
\(11^{\circ }19'\)
\(88^{\circ }41'\)
\(45^{\circ }45'\)
\(54^{\circ }12'\)

9000151306

Level: 
B
Find the angle \(\varphi \) between the lines \(p\) and \(q\) given by their parametric equations. \[ p\colon \begin{aligned}[t] x& = 1 - t, & \\y& = 2 + t,\ t\in \mathbb{R}, \\ \end{aligned}\qquad q\colon \begin{aligned}[t] x& = 4 - k, & \\y& = 5 + k,\ k\in \mathbb{R}. \\ \end{aligned} \]
\(0^{\circ }\)
\(90^{\circ }\)
\(60^{\circ }\)
\(30^{\circ }\)

9000151307

Level: 
B
Find the angle \(\varphi \) between the line \(x + \sqrt{3}y - 6 = 0\) and the line \(p\) given by it's parametric equations. \[ p\colon \begin{aligned}[t] x& = 2 + t,& \\y& = 5,\ t\in \mathbb{R} \\ \end{aligned} \]
\(30^{\circ }\)
\(90^{\circ }\)
\(60^{\circ }\)
\(45^{\circ }\)

9000149410

Level: 
B
Find all lines passing through the point \(A = [-2,-6]\) such that the distance from the point \([0.0]\) to these lines is \(2\sqrt{2}\).
\(p_{1}\colon 7x + y + 20 = 0\), \(p_{2}\colon x - y - 4 = 0\)
\(p\colon 7x - y = 0\)
\(p\colon x + y + 2\sqrt{2} = 0\)
\(p_{1}\colon x - y + 2\sqrt{2} = 0\), \(p_{2}\colon x + y - 2\sqrt{2} = 0\)