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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. - Mathematics

Answer in Brief

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.

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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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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 23 The straight lines
Exercise 23.11 | Q 3 | Page 83
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