The line segment joining the points A(3, 2) and B(5, 1) is divided at the points P in the ratio 1 : 2 and it lies on the line 3x - 18y + k = 0. Find the value of k.

Let (x, y) be the coordinates of the point P which divides the line segment joining A(3, 2) and B(5, 1) in the ratio 1 : 2.

∴ Coordinates of P are $\Big(\dfrac{m$1x2 + m2x1}{m1 + m2}, \dfrac{m1y2 + m2y1}{m1 + m2}\Big).

Putting the values from question in above formula we get,

$\Rightarrow \Big(\dfrac{1 \times 5 + 2 \times 3}{1 + 2}, \dfrac{1 \times 1 + 2 \times 2}{1 + 2}\Big) \\[1em] = \Big(\dfrac{5 + 6}{3}, \dfrac{1 + 4}{3}\Big) \\[1em] = \Big(\dfrac{11}{3}, \dfrac{5}{3}\Big).$

Since, P lies on the line 3x - 18y + k = 0.
∴ It will satisfy it.

$\Rightarrow 3\Big(\dfrac{11}{3}\Big) - 18\Big(\dfrac{5}{3}\Big) + k = 0 \\[1em] \Rightarrow 11 - 30 + k = 0 \\[1em] \Rightarrow -19 + k = 0 \\[1em] \Rightarrow k = 19.$

Hence, the value of k = 19.