In what ratio does the point P(a, -1) divide the join of A(1, -3) and B(6, 2)? Hence, find the value of a.

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

By section-formula,

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

Substituting values we get :

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

By section-formula,

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

Substitute values we get:

$\Rightarrow a = \Big(\dfrac{2 \times 6 + 3 \times 1}{2 + 3}\Big) \\[1em] = \Big(\dfrac{12 + 3}{5}\Big) \\[1em] = \dfrac{15}{5} \\[1em] = 3.$

Hence, point P divide AB in the ratio 2 : 3 and value of a = 3.