Problem Statement
Find the value of $k$ if the points $A(2, 3), B(4, k)$ and $C(6, -3)$ are collinear.
Verified Solution & Marking Scheme
Area of Triangle Formula for Collinear Points
Points are collinear if area of $\triangle ABC = 0$:
$$\frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| = 0$$
$$x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) = 0$$
Substitute Coordinates and Solve
$$2(k - (-3)) + 4(-3 - 3) + 6(3 - k) = 0$$
$$2(k + 3) + 4(-6) + 6(3 - k) = 0$$
$$2k + 6 - 24 + 18 - 6k = 0$$
$$-4k + 0 = 0 \implies -4k = 0 \implies k = 0$$