Find the value of m, if the points (5,1),

Let A ≡ (x1,y1) s (5,1), B = (x2, y2) = (- 2, – 3), C s (x3, y3) = (8,2m)

Since, the points A ≡ (5,1), B ≡ (- 2, – 3) and C ≡ (8,2m) are collinear.

Area of $\triangle A B C=0$

$\Rightarrow \quad \frac{1}{2}\left[x_{1}\left(y_{2}-y_{3}\right)+x_{2}\left(y_{3}-y_{1}\right)+x_{3}\left(y_{1}-y_{2}\right)\right]=0$

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

$\Rightarrow \quad \frac{1}{2}(-15-10 m-4 m+2+32)=0$

$\Rightarrow \quad \frac{1}{2}(-14 m+19)=0 \Rightarrow m=\frac{19}{14}$

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