If the point P(6, -3) lies on the line segment joining points A(4, 2) and B(8, 4), then:

1. AP = $\dfrac{3}{4}$ AB

2. AP = $\dfrac{1}{4}$ AB

3. PB = $\dfrac{1}{3}$ AB

4. AP = $\dfrac{1}{2}$ AB

Let the ratio in which P divides AB be k : 1.

By section-formula,

x = $\dfrac{m$1x2 + m2x1}{m1 + m2}

Substituting values we get :

$\Rightarrow 6 = \Big(\dfrac{k(8) + 1(4)}{k + 1}\Big) \\[1em] \Rightarrow 6 = \Big(\dfrac{8k + 4}{k + 1}\Big) \\[1em] \Rightarrow 6(k + 1) = 8k + 4 \\[1em] \Rightarrow 6k + 6 = 8k + 4 \\[1em] \Rightarrow 6 - 4 = 8k - 6k \\[1em] \Rightarrow 2 = 2k \\[1em] \Rightarrow k = \dfrac{1}{1} = 1:1.$

This means that P is the midpoint of AB.

∴ AP = $\dfrac{1}{2}$ AB.

Hence, Option 4 is the correct option.