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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 $\left(p_{3}, q_{3}\right)$
Options:
$\left(p_{2}, q_{2}\right)$ and $\left(p_{3}, q_{3}\right)$
Solution:
2475 Upvotes
Verified Answer
The correct answer is:
are collinear
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.
$$
\begin{array}{l}
\mathrm{p}_{1} \mathrm{x}+\mathrm{q}_{1} \mathrm{y}-1=0 \\
\mathrm{p}_{2} \mathrm{x}+\mathrm{q}_{2} \mathrm{y}-1=0
\end{array}
$$
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.
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