Question

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.

Answer 1

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.

\[\therefore \begin{vmatrix}p_1 & q_1 & - 1 \\ p_2 & q_2 & - 1 \\ p_3 & q_3 & - 1\end{vmatrix} = 0\]

\[ \Rightarrow - \begin{vmatrix}p_1 & q_1 & 1 \\ p_2 & q_2 & 1 \\ p_3 & q_3 & 1\end{vmatrix} = 0\]

\[ \Rightarrow \begin{vmatrix}p_1 & q_1 & 1 \\ p_2 & q_2 & 1 \\ p_3 & q_3 & 1\end{vmatrix} = 0\]

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