Question

If the points (5, 1), (–2, –3) and (8, 2m) are collinear, then m = ___________.

Answer 1

If the area of the triangle formed by three points is equal to zero, then the points are collinear.

Area of the triangle formed by the vertices $\left(x_1,y_1\right),\left(x_2,y_2\right)\mathrm{and}\left(x_3,y_3\right)\mathrm{is}$ $\frac{1}{2}\left|x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)\right|$.

Now, the given points (5, 1), (–2, –3) and (8, 2m) are collinear.

Therefore, Area of triangle formed by them is equal to zero.

$$\begin{gathered} \mathrm{Area}\mathrm{of}\mathrm{triangle}=0 \\ \Rightarrow \frac{1}{2}\left|5\left(-3-2m\right)+\left(-2\right)\left(2m-1\right)+8\left(1-\left(-3\right)\right)\right|=0 \\ \Rightarrow \frac{1}{2}\left|-15-10m-4m+2+8\left(1+3\right)\right|=0 \\ \Rightarrow \left|-13-14m+8\left(4\right)\right|=0 \\ \Rightarrow \left|-13-14m+32\right|=0 \\ \Rightarrow \left|-14m+19\right|=0 \\ \Rightarrow 19-14m=0 \\ \Rightarrow 14m=19 \\ \Rightarrow m=\frac{19}{14} \end{gathered}$$

Hence, m = $\underline{\frac{19}{14}}.$