Question

If the points A(2, 3), B(4, k) and C(6, −3) are collinear, find the value of k.

Answer 1

The given points are $A\left(2,3\right),B\left(4,k\right)\text{and}C\left(6,-3\right)$.
Here, $\left(x_1=2,y_1=3\right),\left(x_2=4,y_2=k\right)\text{and}\left(x_3=6,y_3=-3\right)$
Points A,B and C are collinear. Then,
$$\begin{gathered} x_1\left(y_2-y_3\right)+x_2\left(y_3-y_1\right)+x_3\left(y_1-y_2\right)=0 \\ \Rightarrow 2\left(k+3\right)+4\left(-3-3\right)+6\left(3-k\right)=0 \\ \Rightarrow 2k+6-24+18-6k=0 \\ \Rightarrow -4k=0 \\ \Rightarrow k=0 \end{gathered}$$