Find the value of $ m $ if the points $ (5,1),(-2,-3) $ and $ (8,2 m) $ are collinear.

Given: 

Points \( (5,1),(-2,-3) \) and \( (8,2 m) \) are collinear.

To do: 

We have 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.

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

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

$\Rightarrow \frac{1}{2}[-2+4m+5( 2m+3)+8(-4)]=0$

$\Rightarrow (-2+4m+10m+15-32)=0$

$\Rightarrow (14m-19)=0$

$\Rightarrow 14m=19$

$\Rightarrow m=\frac{19}{14}$

Therefore, the value of $m$ is $\frac{19}{14}$.