Analytic geometry in a plane

9000090906

Level: 
C
Given lines \(p\) and \(q\), find \(m\in \mathbb{R}\) such that the lines \(p\) and \(q\) are parallel. \[ \begin{aligned}p\colon x& = 1 + t, & \\y & = -3t;\ t\in \mathbb{R}, \\ \end{aligned}\qquad \begin{aligned}q\colon x& = 3 - 2u, & \\y & = 1 + mu;\ u\in \mathbb{R} \\ \end{aligned} \]
\(m = 6\)
\(m = \frac{3} {2}\)
\(m = -\frac{2} {3}\)
does not exist

9000090907

Level: 
C
Given points \(A = [2;m]\) and \(B = [-1;0]\), find \(m\in \mathbb{R}\) such that the line \(p\) is parallel to the line passes through the points \(A\), \(B\). \[ \begin{aligned}p\colon x& = 3 + 2t, & \\y & = 5 - t;\ t\in \mathbb{R} \\ \end{aligned} \]
\(m = -\frac{3} {2}\)
\(m = \frac{3} {2}\)
\(m = -\frac{2} {3}\)
\(m = 2\)
does not exist

9000090908

Level: 
C
Given points \(A = [2;1]\) and \(B = [m;0]\), find \(m\in \mathbb{R}\) such that the line \(p\) is parallel to the line passes through the points \(A\), \(B\). \[ p\colon 3x - y + 17 = 0 \]
\(m = \frac{5} {3}\)
\(m = 4\)
\(m = \frac{5} {2}\)
\(m = -1\)
another solution

9000090909

Level: 
C
Given lines \(p\) and \(q\), find \(m\in \mathbb{R}\) such that the lines \(p\) and \(q\) are parallel. \[ p\colon 2x+my-3 = 0,\qquad \begin{aligned}[t] q\colon x& = 1 + t, & \\y & = 2 - t;\ t\in \mathbb{R} \\ \end{aligned} \]
\(m = 2\)
\(m = -2\)
\(m = 11\)
\(m = -\frac{1} {11}\)
does not exist

9000090910

Level: 
C
Given lines \(p\) and \(q\), find \(m\in \mathbb{R}\) such that the line \(p\) is parallel to \(q\). \[ p\colon x+4y-3 = 0,\qquad \begin{aligned}[t] q\colon x& = 1 + mt,& \\y & = 2 - 3t;\ t\in \mathbb{R} \\ \end{aligned} \]
\(m = 12\)
\(m = -\frac{1} {12}\)
\(m = 4\)
\(m = \frac{5} {2}\)
\(m = -1\)

9000106805

Level: 
C
Given points \(A = [0;5]\), \(B = [6;1]\), \(C = [7;9]\), find the direction vector of the line passing through the point \(A\) and the midpoint of the segment \(BC\) (i.e. the median of the triangle \(ABC\) through the vertex \(A\)).
\((1;0)\)
\((1;8)\)
\((1;9)\)
\((6.5;5)\)

9000106806

Level: 
C
Given points \(A = [0;5]\), \(B = [6;1]\), \(C = [7;9]\), find the direction vector of the line passing through the point \(A\) and perpendicular to the segment \(BC\) (i.e. the line which contains the altitude of the triangle \(ABC\) through the point \(A\)).
\((8;-1)\)
\((1;8)\)
\((1;9)\)
\((-9;1)\)

9000106807

Level: 
C
Consider the points \(A = [0;5]\), \(B = [6;1]\), \(C = [7;9]\) and the triangle \(ABC\). Find the direction vector of the line which is the perpendicular bisector of the side \(b\) (i.e. the line through the midpoint of the side \(AC\) which is perpendicular to the segment \(AC\)).
\((4;-7)\)
\((7;4)\)
\((7;9)\)
\((7;-9)\)

9000106808

Level: 
C
Consider the points \(A = [0;5]\), \(B = [6;1]\), \(C = [7;9]\) and the triangle \(ABC\). Find the direction vector of the line which is the bisector of the angle \(ACB\) (i.e. the line which splits the internal angle at the point \(C\) into two angles with equal measures).
\((2;3)\)
\((6;-4)\)
\((7;9)\)
\((7;8)\)