If the points $( -2,\ 3),\ ( 5,\ 1),\ ( 8,\ 2m)$ are collinear then find the value of $m$.

Given: Points $( -2,\ 3),\ ( 5,\ 1),\ ( 8,\ 2m)$ are collinear.

To do: To find the value of $m$.

Solution:

Given points are: $( -2,\ 3),\ ( 5,\ 1),\ ( 8,\ 2m)$.

Here $x_1=-2,\ y_1=3,\ x_2=5,\ y_2=1,\ x_3=8,\ y_3=2m$

If given points are collinear, then area of the triangle formed by the given points is zero.

$\Rightarrow \frac{1}{2}[x_1( y_2-y_3)+x_2( y_3-y_1)+x_3( y_1-y_2)]=0$

$\Rightarrow \frac{1}{2}[-2( 1-3)+5( 2m-(-2))+8(3-1)]=0$

$\Rightarrow \frac{1}{2}[-2(-2)+5( 2m+2)+8(2)]=0$

$\Rightarrow \frac{1}{2}[-4+10m+10+16]=0$

$\Rightarrow \frac{1}{2}[22+10m]=0$

$\Rightarrow 22+10m=0$

$\Rightarrow 10m=-22$

$\Rightarrow m=\frac{-22}{10}=-\frac{11}{5}$