In what ratio does the point P(2, -5) divide the join of A(-3, 5) and B(4, -9)?

Let the point P(2, -5) divide the line segment AB in the ratio k : 1.

By section-formula,

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

Substituting values we get :

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

Hence, point P divide AB in the ratio 5 : 2.