Question

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

Answer 1

Find the value .

For, collinear points area of triangle$=0$

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

Area of triangle $\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{}\mathbf{\left[}{\mathit{x}}_{\mathbf{1}\mathbf{}}\mathbf{\right(}{\mathit{y}}_{\mathbf{2}}\mathbf{}\mathbf{–}\mathbf{}{\mathit{y}}_{\mathbf{3}}\mathbf{)}\mathbf{}\mathbf{+}\mathbf{}{\mathit{x}}_{\mathbf{2}}\mathbf{}\mathbf{(}{\mathit{y}}_{\mathbf{3}}\mathbf{–}{\mathit{y}}_{\mathbf{1}\mathbf{}}\mathbf{)}\mathbf{}\mathbf{+}\mathbf{}{\mathit{x}}_{\mathbf{3}}\mathbf{(}{\mathit{y}}_{\mathbf{1}}\mathbf{}\mathbf{–}\mathbf{}{\mathit{y}}_{\mathbf{2}}\mathbf{\left)}\mathbf{\right]}\mathbf{=}\mathbf{0}$

$$\begin{gathered} \Rightarrow \frac{1}{2}\left[5\right(\left(-3\right)–2m)+(–2\left)\right(2m–1)+8(1–(–3)\left)\right]=0 \\ \Rightarrow -15–10m–4m+2+32=0 \\ \Rightarrow -14m+19=0 \\ \Rightarrow m=\frac{19}{14} \end{gathered}$$

Hence, the required answer is $\frac{19}{14}$.