If the lines $\mathrm{p}_{1} \mathrm{x}+\mathrm{q}_{1} \mathrm{y}=1, \mathrm{p}_{2}…
If the lines $\mathrm{p}_{1} \mathrm{x}+\mathrm{q}_{1} \mathrm{y}=1, \mathrm{p}_{2} \mathrm{x}+\mathrm{q}_{2} \mathrm{y}=1$ and $\mathrm{p}_{3} \mathrm{x}+ \mathrm{q}_{3} \mathrm{y}=1$ be concurrent, then the points $\left(\mathrm{p}_{1}, \mathrm{q}_{1}\right),$
$\left(p_{2}, q_{2}\right)$ and $\left(p_{3}, q_{3}\right)$
are collinear
form an equilateral triangle
form a sealene triangle
form a right angled triangle
Solution
The equations of the lines are
$
\begin{array}{l}
\mathrm{p}_{1} \mathrm{x}+\mathrm{q}_{1} \mathrm{y}-1=0 \\
\mathrm{p}_{2} \mathrm{x}+\mathrm{q}_{2} \mathrm{y}-1=0
\end{array}
$
and $\mathrm{p}_{3} \mathrm{x}+\mathrm{q}_{3} \mathrm{y}-1=0$
As they are concurrent,
$
\left|\begin{array}{lll}
\mathrm{p}_{1} & \mathrm{q}_{1} & -1 \\
\mathrm{p}_{2} & \mathrm{q}_{2} & -1 \\
\mathrm{p}_{3} & \mathrm{q}_{3} & -1
\end{array}\right|=0 \Rightarrow\left|\begin{array}{lll}
\mathrm{p}_{1} & \mathrm{q}_{1} & 1 \\
\mathrm{p}_{2} & \mathrm{q}_{2} & 1 \\
\mathrm{p}_{3} & \mathrm{q}_{3} & 1
\end{array}\right|=0
$
This is also the condition for the points $\left(\mathrm{p}_{1}, \mathrm{q}_{1}\right),\left(\mathrm{p}_{2}, \mathrm{q}_{2}\right)$ and $\left(\mathrm{p}_{3}, \mathrm{q}_{3}\right)$ to be collinear