Analytical plane geometry

Relative Position of Points and Lines in a Plane

Submitted by ladislav.foltyn on Fri, 05/31/2019 - 12:44
Question: 
Suppose a line $p$ passes through the point $A=[2;3]$ and additionally has the property described in the first column of the table. In each row mark the line’s equation. \begin{align*} p_1\colon y&=-3x+9 & p_2\colon y&=x+1 & p_3\colon y&=2x-1 \\ p_4\colon y&=-2x+7 & p_5\colon y&=3 & p_6\colon x&=2 \end{align*}

Parametric Equations of a Line and of Objects on a Line

Submitted by ladislav.foltyn on Thu, 05/02/2019 - 14:40
Question: 
\vspace*{-1em} We are given points $A=[-1;3]$ and $B=[3;6]$. Further, we are given parametric equations of a straight line $AB$: $x=-1+4t$, $y=3+3t;$ $t\in\mathbb{R}$. Match each described object lying on the straight line $AB$ with the corresponding set of values of parameter $t$. \vspace*{-1em}

Parallel Lines

Submitted by ladislav.foltyn on Thu, 05/02/2019 - 14:03
Question: 
In the table, mark a cell, if the two corresponding lines are parallel to each other. \begin{align*} a\colon&\, \left\{\begin{array}{ll} x=3+t\text{, } & \\ y=-3-t; & t\in\mathbb{R}\end{array}\right. & b\colon&\, y=3x-2 & c\colon&\, 4x-2y+5=0 \\ d\colon&\, y=\frac23x-7 & e\colon&\, 2x+y-6=0 & f\colon&\, \left\{\begin{array}{ll} x=3+4t\text{, } & \\ y=\phantom{3\,}-6t; & t\in\mathbb{R}\end{array}\right. \end{align*}