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Question
Find the conditions that the straight lines y = m1 x + c1, y = m2 x + c2 and y = m3 x + c3 may meet in a point.
Solution
The given lines can be written as follows:
\[m_1 x - y + c_1 = 0\] ... (1)
\[m_2 x - y + c_2 = 0\] ... (2)
\[m_3 x - y + c_3 = 0\] ... (3)
It is given that the three lines are concurrent.
\[\therefore \begin{vmatrix}m_1 & - 1 & c_1 \\ m_2 & - 1 & c_2 \\ m_3 & - 1 & c_3\end{vmatrix} = 0\]
\[ \Rightarrow m_1 \left( - c_3 + c_2 \right) + 1\left( m_2 c_3 - m_3 c_2 \right) + c_1 \left( - m_2 + m_3 \right) = 0\]
\[ \Rightarrow m_1 \left( c_2 - c_3 \right) + m_2 \left( c_3 - c_1 \right) + m_3 \left( c_1 - c_2 \right) = 0\]
Hence, the required condition is \[m_1 \left( c_2 - c_3 \right) + m_2 \left( c_3 - c_1 \right) + m_3 \left( c_1 - c_2 \right) = 0\].
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