If three lines whose equations are $y = m_{1} x + c_{1}, y = m_{2} x + c_{2}$ and $y = m_{3} + c_{3}$ are concurrent, then show that $m_{1} (c_{2} - c_{3}) + m_{2} (c_{3} - c_{1}) + m_{3} (c_{1} - c_{2}) = 0$

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Solution

The equations of the given lines are $y = m_{1} x + c_{1}$ ...(1)
$y = m_{2} x + c_{2}$ ...(2)
$y = m_{3} x + c_{3}$ ....(3)
On subtracting equation (1) from (2), we obtain
$0 = (m_{2} - m_{1}) x + (c_{2} - c_{1})$
$\Rightarrow (m_{1} - m_{2}) x = c_{2} - c_{1}$
$\Rightarrow x = \frac{c_{2} - c_{1}}{m_{1} - m_{2}}$
On substituting this value of x in (1), we obtain
$y = m_{1} {\frac{c_{2} - c_{1}}{m_{1} - m_{2}}} + c_{1}$
$y = \frac{m_{1} c_{2} - m_{1} c_{1}}{m_{1} - m_{2}} + c_{1}$
$y = \frac{m_{1} c_{2} - m_{1} c_{1} + m_{1} c_{1} - m_{2} c_{1}}{m_{1} - m_{2}}$
$y = \frac{m_{1} c_{2} - m_{2} c_{1}}{m_{1} - m_{2}}$
$∴ {\frac{c_{2} - c_{1}}{m_{1} - m_{2}}, \frac{m_{1} c_{2} - m_{2} c_{1}}{m_{1} - m_{2}}}$ is the point of intersection of lines (1) and (2)
It is given that lines (1),(2), and (3) are concurrent.
Hence, the point of intersection of lines (1) and (2) will also satisfy equation (3).
$\frac{m_{1} c_{2} - m_{2} c_{1}}{m_{1} - m_{2}} = m_{3} {\frac{c_{2} - c_{1}}{m_{1} - m_{2}}} + c_{3}$
$\frac{m_{1} c_{2} - m_{2} c_{1}}{m_{1} - m_{2}} = \frac{m_{3} c_{2} - m_{3} c_{1} + c_{3} m_{1} - c_{3} m_{2}}{m_{1} - m_{2}}$
$m_{1} (c_{2} - c_{3}) + m_{2} (c_{3} - c_{1}) + m_{3} (c_{1} - c_{2}) = 0$
Hence, $m_{1} (c_{2} - c_{3}) + m_{2} (c_{3} - c_{1}) + m_{3} (c_{1} - c_{2}) = 0$

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Similar questions

y = m_{1} x + c_{1}, y = m_{2} x + c_{2}

y = m_{3} x + c_{3}

m_{1} (c_{2} - c_{1}) + m_{2} (c_{3} - c_{1}) + m_{3} (c_{1} - c_{2})

If the three lines $y = m_{1} x + c_{1}, y = m_{2} x + c_{2}$ and $y = m_{3} x + c_{3}$ are concurrent then show that,

$m_{1} (c_{2} - c_{3}) + m_{2} (c_{3} - c_{1}) + m_{3} (c_{1} - c_{2}) = 0$

y = m_{1} x + c_{1}, y = m_{2} x + c_{2}

y = m_{2} x + c_{2}

m_{1} (c_{2} - c_{1}) + m_{2} (c_{3} - c_{1}) + m_{3} (c_{1} - c_{2}) = 0

y = m_{1} x + c_{1}

y = m_{2} x + c_{2}

y = m_{3} x + c_{3}

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.

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