In what ratio does the point P(a, 2) divide the line segment joining the points A(5, -3) and B(-9, 4) ? Also, find the value of 'a'.

Let ratio be k : 1.

By section formula,

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

Substituting values for y-coordinate we get :

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

k : 1 = $\dfrac{5}{2} : 1$ = 5 : 2.

So, m₁ : m₂ = 5 : 2.

Substituting values for x-coordinate we get :

$\Rightarrow a = \dfrac{5 \times -9 + 2 \times 5}{5 + 2} \\[1em] \Rightarrow a = \dfrac{-45 + 10}{7} \\[1em] \Rightarrow a = \dfrac{-35}{7} = -5.$

Hence, a = -5 and ratio = 5 : 2.