If the lines p₁ x + q₁ y = 1, p₂ x + q₂ y = 1 and p₃ x + q₃ y = 1 be concurrent, show that the points (p₁, q₁), (p₂, q₂) and (p₃, q₃) are collinear.

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Solution

The given lines can be written as follows:

p₁ x + q₁ y $-$ 1 = 0 ... (1)

p₂ x + q₂ y $-$ 1 = 0 ... (2)

p₃ x + q₃ y $-$ 1 = 0 ... (3)

It is given that the three lines are concurrent.

$∴ |\begin{matrix} p_{1} & q_{1} & - 1 \\ p_{2} & q_{2} & - 1 \\ p_{3} & q_{3} & - 1 \end{matrix}| = 0 \Rightarrow - |\begin{matrix} p_{1} & q_{1} & 1 \\ p_{2} & q_{2} & 1 \\ p_{3} & q_{3} & 1 \end{matrix}| = 0 \Rightarrow |\begin{matrix} p_{1} & q_{1} & 1 \\ p_{2} & q_{2} & 1 \\ p_{3} & q_{3} & 1 \end{matrix}| = 0$

This is the condition for the collinearity of the three points, (p₁, q₁), (p₂, q₂) and (p₃, q₃).

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