Find the value of k, if the points *A*(7, −2), *B* (5, 1) and *C *(3, 2*k*) are collinear.

#### 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.

The three given points are *A*(7*, −*2)*, B*(5*,* 1) and *C*(3*,* 2*k*). It is also said that they are collinear and hence the area enclosed by them should be 0.

\[∆ = \frac{1}{2}\left| \left( 7 \times 1 + 5 \times 2k + 3 \times - 2 \right) - \left( 5 \times - 2 + 3 \times 1 + 7 \times 2k \right) \right|\]

\[ 0 = \frac{1}{2}\left| \left( 7 + 10k - 6 \right) - \left( - 10 + 3 + 14k \right) \right|\]

\[ 0 = \frac{1}{2}\left| - 4k + 8 \right|\]

\[ 0 = - 4k + 8\]

\[ k = 2\]

Hence the value of ‘*k*’ for which the given points are collinear is **k = 2** .