Find the conditions that the straight lines y = m_{1} x + c_{1}, y = m_{2} x + c_{2} and y = m_{3} x + c_{3} 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 \[m_1 \left( c_2 - c_3 \right) + m_2 \left( c_3 - c_1 \right) + m_3 \left( c_1 - c_2 \right) = 0\].