Find the ratio in which the point P(-3, p) divides the line segment joining the points (-5, -4) and (-2, 3). Hence, find the value of p.

Let (-3, p) divides the line segment in the ratio of m : n.

By section-formula,

x-coordinate = $\Big(\dfrac{mx$2 + nx1}{m + n}\Big)

$\therefore -3 = \dfrac{m \times (-2) + n \times (-5)}{m + n} \\[1em] \Rightarrow -3 = \dfrac{-2m - 5n}{m + n} \\[1em] \Rightarrow -3(m + n) = -2m - 5n \\[1em] \Rightarrow -3m - 3n = -2m - 5n \\[1em] \Rightarrow -3m + 2m = -5n + 3n \\[1em] \Rightarrow -m = -2n \\[1em] \Rightarrow \dfrac{m}{n} = \dfrac{2}{1} \\[1em] \Rightarrow m : n = 2 : 1. \\[1em]$

By section formula, y-coordinate = p

$= \dfrac{my$2 + ny1}{m + n} \\[1em] = \dfrac{2 \times 3 + 1 \times (-4)}{2 + 1} \\[1em] = \dfrac{6 - 4}{3} \\[1em] = \dfrac{2}{3}.

Hence, the value of p = $\dfrac{2}{3}$ and the ratio in which point P divides the line segment is 2 : 1.