Determine the ratio in which the line y = 2 + 3x divides the line segment AB joining the points A(-3, 9) and B(4, 2).

By two point form,

Equation of line :

⇒ y - y₁ = $\dfrac{y$2 - y1}{x2 - x1}(x - x₁)

Equation of AB :

$\Rightarrow y - 9 = \dfrac{2 - 9}{4 - (-3)}[x - (-3)] \\[1em] \Rightarrow y - 9 = \dfrac{-7}{4 + 3}[x + 3] \\[1em] \Rightarrow y - 9 = \dfrac{-7}{7}[x + 3] \\[1em] \Rightarrow y - 9 = -1[x + 3] \\[1em] \Rightarrow y - 9 = -x - 3 \\[1em] \Rightarrow y + x = -3 + 9 \\[1em] \Rightarrow x + y = 6.$

Solving equation y = 2 + 3x and x + y = 6 simultaneously,

⇒ x + y = 6 …….(1)

⇒ y = 2 + 3x …….(2)

Substituting value of y from equation (2) in (1), we get :

⇒ x + (2 + 3x) = 6

⇒ 4x + 2 = 6

⇒ 4x = 6 - 2

⇒ 4x = 4

⇒ x = $\dfrac{4}{4}$

⇒ x = 1.

Substituting value of x in equation (2), we get :

⇒ y = 2 + 3(1) = 2 + 3 = 5.

Let (1, 5) divide the line AB in the ratio k : 1.

By section formula,

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

Substituting values we get :

$\Rightarrow (1, 5) = \Big(\dfrac{k \times 4 + 1 \times -3}{k + 1}, \dfrac{k \times 2 + 1 \times 9}{k + 1}\Big) \\[1em] \Rightarrow (1, 5) = \Big(\dfrac{4k - 3}{k + 1}, \dfrac{2k + 9}{k + 1}\Big) \\[1em] \Rightarrow 1 = \dfrac{4k - 3}{k + 1} \text{ or } 5 = \dfrac{2k + 9}{k + 1} \\[1em] \Rightarrow 1(k + 1) = 4k - 3 \text{ or } 5(k + 1) = 2k + 9 \\[1em] \Rightarrow k + 1 = 4k - 3 \text{ or } 5k + 5 = 2k + 9 \\[1em] \Rightarrow 4k - k = 1 + 3 \text{ or } 5k - 2k = 9 - 5 \\[1em] \Rightarrow 3k = 4 \text{ or } 3k = 4 \\[1em] \Rightarrow k = \dfrac{4}{3} \\[1em] \Rightarrow k : 1 = \dfrac{4}{3} : 1 = 4 : 3.$

Hence, the line y = 2 + 3x divides the line segment AB in the ratio 4 : 3.