If the lines $p_{1} x + q_{1} y = 1, p_{2} x + q_{2} y = 1$ and $p_{3} x + q_{3} y = 1$ be concurrent, then the points $(p_{1}, q_{1}) (p_{2}, q_{2}) (p_{3}, q_{3})$

Are collinear

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Solution

The correct option is C Are collinear
The equations of the lines are
$p_{1} x + q_{1} y - 1 = 0$ .....(i)
$p_{2} x + q_{2} y - 1 = 0$ .....(ii)
$p_{3} x + q_{3} y - 1 = 0$ ...(iii)
They are concurrent.
Therefore, $∣ ∣ ∣ ∣ ∣ \begin{matrix} p_{1} & q_{1} & - 1 p_{2} & q_{2} & - 1 p_{3} & q_{3} & - 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0$
$\Rightarrow ∣ ∣ ∣ ∣ ∣ \begin{matrix} p_{1} & q_{1} & 1 p_{2} & q_{2} & 1 p_{3} & q_{3} & 1 \end{matrix} ∣ ∣ ∣ ∣ ∣ = 0$
This is also the condition for the points $(p_{1}, q_{1}), (p_{2}, q_{2}), (p_{3}, q_{3})$ to be collinear.

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