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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.
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.
\[\therefore \begin{vmatrix}p_1 & q_1 & - 1 \\ p_2 & q_2 & - 1 \\ p_3 & q_3 & - 1\end{vmatrix} = 0\]
\[ \Rightarrow - \begin{vmatrix}p_1 & q_1 & 1 \\ p_2 & q_2 & 1 \\ p_3 & q_3 & 1\end{vmatrix} = 0\]
\[ \Rightarrow \begin{vmatrix}p_1 & q_1 & 1 \\ p_2 & q_2 & 1 \\ p_3 & q_3 & 1\end{vmatrix} = 0\]
This is the condition for the collinearity of the three points, (p1, q1), (p2, q2) and (p3, q3).
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