In what ratio does the point (1, a) divide the join of (-1, 4) and (4, -1) ?

Also, find the value of a.

We know that,

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

Substituting values we get,

$\Rightarrow 1 = \dfrac{m$1 \times 4 + m2 \times -1}{m1 + m2} \\[1em] \Rightarrow m1 + m2 = 4m1 - m2 \\[1em] \Rightarrow m2 + m2 = 4m1 - m1 \\[1em] \Rightarrow 2m2 = 3m1 \\[1em] \Rightarrow \dfrac{m1}{m2} = \dfrac{2}{3}.

∴ m₁ : m₂ = 2 : 3.

We know that,

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

Substituting values we get,

$\Rightarrow a = \dfrac{2 \times -1 + 3 \times 4}{2 + 3} \\[1em] \Rightarrow a = \dfrac{-2 + 12}{5} \\[1em] \Rightarrow a = \dfrac{10}{5} \\[1em] \Rightarrow a = 2.$

Hence, ratio = 2 : 3 and a = 2.