Question

Find the value of k if (k, 3), (2, $-$4) and ($-$k + 1, $-$2) are collinear points .

Answer 1

Given: Points (k, 3), (2, $-$4) and ($-$k + 1, $-$2) are collinear.
So, the slope of the line joining (k, 3) and (2, $-$4) = slope of the line joining (2, $-$4) and ($-$k+1,$-$2)

$$\begin{gathered} \mathrm{Now},\frac{y_2-y_1}{x_2-x_1}=\frac{y_3-y_2}{x_3-x_3} \\ \Rightarrow \frac{-4-3}{2-k}=\frac{-2-(-4)}{-k+1-2} \\ \Rightarrow \frac{-7}{2-k}=\frac{2}{-k-1} \\ \Rightarrow -7\left(-k-1\right)=2(2-k) \\ \Rightarrow 7k+7=4-2k \\ \Rightarrow 7k+2k=4-7 \\ \Rightarrow 9k=-3 \\ \Rightarrow k=\frac{-3}{9}=\frac{-1}{3} \end{gathered}$$