Line 3x + 4y = 24 meets x-axis at point A and y-axis at point B. If the point P divides line segment AB internally in the ratio 2 : 1, the co-ordinates of point P are :

1. (3, 4)

2. (4, 3)

3. $\Big(4, 2\dfrac{2}{3}\Big)$

4. $\Big(2\dfrac{2}{3}, 4\Big)$

We know that,

y-coordinate of any point at x-axis is 0.

Substituting y = 0 in 3x + 4y = 24,

⇒ 3x + 4(0) = 24

⇒ 3x = 24

⇒ x = $\dfrac{24}{3}$ = 8.

A = (8, 0).

x-coordinate of any point at y-axis is 0.

Substituting x = 0 in 3x + 4y = 24,

⇒ 3(0) + 4y = 24

⇒ 4y = 24

⇒ y = $\dfrac{24}{4}$ = 6.

B = (0, 6).

By section formula,

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

Given, P divides line AB internally in the ratio 2 : 1.

Substituting values we get :

$\Rightarrow (x, y) = \Big(\dfrac{2 \times 0 + 1 \times 8}{2 + 1}, \dfrac{2 \times 6 + 1 \times 0}{2 + 1}\Big) \\[1em] \Rightarrow (x, y) = \Big(\dfrac{0 + 8}{3}, \dfrac{12 + 0}{3}\Big) \\[1em] \Rightarrow (x, y) = \Big(\dfrac{8}{3}, \dfrac{12}{3}\Big) \\[1em] \Rightarrow (x, y) = \Big(2\dfrac{2}{3}, 4\Big).$

Hence, Option 4 is the correct option.