In what ratio does the point (a, 6) divide the join of (-4, 3) and (2, 8) ?

Also, find the value of a.

Let ratio in which point (a, 6) divide the join of (-4, 3) and (2, 8) be m₁ : m₂.

By section formula,

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

Substituting values we get,

$\Rightarrow 6 = \dfrac{m$1 \times 8 + m2 \times 3}{m1 + m2} \\[1em] \Rightarrow 6m1 + 6m2 = 8m1 + 3m2 \\[1em] \Rightarrow 8m1 - 6m1 = 6m2 - 3m2 \\[1em] \Rightarrow 2m1 = 3m2 \\[1em] \Rightarrow \dfrac{m1}{m2} = \dfrac{3}{2}.

∴ m₁ : m₂ = 3 : 2.

We know that,

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

Substituting values we get,

$\Rightarrow a = \dfrac{3 \times 2 + 2 \times -4}{3 + 2} \\[1em] \Rightarrow a = \dfrac{6 - 8}{5} \\[1em] \Rightarrow a = -\dfrac{2}{5}.$

Hence, ratio = 3 : 2 and a = $-\dfrac{2}{5}$.