If the point P (*m*, 3) lies on the line segment joining the points \[A\left( - \frac{2}{5}, 6 \right)\] and *B* (2, 8), find the value of *m*.

#### Solution

The formula for the area ‘*A*’ encompassed by three points ( x_{1 }, y_{1}) , (x_{2} , y_{2}) and (x_{3} , y_{3}) is given by the formula,

\[∆ = \frac{1}{2}\left| \left( x_1 y_2 + x_2 y_3 + x_3 y_1 \right) - \left( x_2 y_1 + x_3 y_2 + x_1 y_3 \right) \right|\]

If three points are collinear the area encompassed by them is equal to 0.

It is said that the point *P*(*m,*3) lies on the line segment joining the points* A`(-2/5,6)` *and *B*(2*,*8). Hence we understand that these three points are collinear. So the area enclosed by them should be 0.

\[∆ = \frac{1}{2}\left| \left( - \frac{2}{5} \times 3 + m \times 8 + 2 \times 6 \right) - \left( m \times 6 + 2 \times 3 + \left( - \frac{2}{5} \right) \times 8 \right) \right|\]

\[ 0 = \frac{1}{2}\left| \left( - \frac{6}{5} + 8m + 12 \right) - \left( 6m + 6 - \frac{16}{5} \right) \right|\]

\[ 0 = \frac{1}{2}\left| 2m + 8 \right|\]

\[ 0 = 2m + 8\]

\[ m = - 4\]

Hence the value of ‘*m*’ for which the given condition is satisfied is m = - 4 .