Question

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

Answer 1

Points A, B, C will be collinear if the area of $\Delta ABC=0$.
i.e., $\frac{1}{2}\left[x_1\right(y_2-y_3)+x_2[y_3-y_1)+x_3(y_1-y_2\left)\right]=0$

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

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

$\Rightarrow -14m-15+34=0$

$\Rightarrow -14m+19=0$

$\Rightarrow -14m=-19$

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

Hence, the rquired value of $m=\frac{19}{14}$.