If the point R(k, 4) divides the line segment joining the points P(2, 6) and Q(5, 1) in the ratio 2 : 3, then the value of k is:

1. -5

2. $\Big(\dfrac{-16}{5}\Big)$

3. 5

4. $\Big(\dfrac{16}{5}\Big)$

Given,

R = (k, 4)

m₁ : m₂ = 2 : 3

R(k, 4) divides the line segment joining the points P(2, 6) and Q(5, 1) in the ratio 2 : 3.

By section-formula,

(x, y) = $\Big(\dfrac{m$1x2 + m2x1}{m1 + m2}, \dfrac{m1y2 + m2y1}{m1 + m2}\Big)

Substituting values we get :

$\Rightarrow (k, 4) = \Big(\dfrac{2 \times 5 + 3 \times 2}{2 + 3}, \dfrac{2 \times 1 + 3 \times 6}{2 + 3}\Big) \\[1em] \Rightarrow (k, 4) = \Big(\dfrac{10 + 6}{5}, \dfrac{2 + 18}{5}\Big) \\[1em] \Rightarrow (k, 4) = \Big(\dfrac{16}{5}, \dfrac{20}{5}\Big) \\[1em] \Rightarrow (k, 4) = \Big(\dfrac{16}{5}, 4\Big).$

Thus, k = $\dfrac{16}{5}$.

Hence, Option 4 is the correct option.