P(1, -2) is a point on the line segment A(3, -6) and B(x, y) such that AP : PB is equal to 2 : 3. Find the co-ordinates of B.

Given,

m₁ : m₂ = 2 : 3

By section-formula,

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

Substituting values we get :

$\Rightarrow 1 = \Big(\dfrac{2 \times x + 3 \times 3}{2 + 3}\Big) \\[1em] \Rightarrow 1 = \Big(\dfrac{2x + 9}{5}\Big) \\[1em] \Rightarrow 5 = 2x + 9 \\[1em] \Rightarrow 5 - 9 = 2x \\[1em] \Rightarrow -4 = 2x \\[1em] \Rightarrow x = \dfrac{-4}{2} \\[1em] \Rightarrow x = -2.$

By section-formula,

y = $\Big(\dfrac{m$1y2 + m2y1}{m1 + m2}\Big)

Substituting values we get :

$\Rightarrow -2 = \Big(\dfrac{2 \times y + 3 \times -6}{2 + 3}\Big) \\[1em] \Rightarrow -2 = \Big(\dfrac{2y - 18}{5}\Big) \\[1em] \Rightarrow -10 = 2y - 18 \\[1em] \Rightarrow -10 + 18 = 2y \\[1em] \Rightarrow 8 = 2y \\[1em] \Rightarrow y = \dfrac{8}{2} \\[1em] \Rightarrow y = 4.$

Hence, the coordinates of B are (-2, 4).