The line segment joining the points M(5, 7) and N(-3, 2) is intersected by the y-axis at point L. Write down the abscissa of L. Hence, find the ratio in which L divides MN. Also, find the co-ordinates of L.

Since, L lies on y-axis let its co-ordinates be (0, y).

Let L divide MN in ratio m₁ : m₂.

By formula,

$\Rightarrow x = \dfrac{m$1x2 + m2x1}{m1 + m2} \\[1em] \Rightarrow 0 = \dfrac{m1 \times -3 + m2 \times 5}{m1 + m2} \\[1em] \Rightarrow 0 = -3m1 + 5m2 \\[1em] \Rightarrow 3m1 = 5m2 \\[1em] \Rightarrow \dfrac{m1}{m2} = \dfrac{5}{3}.

m₁ : m₂ = 5 : 3.

By formula,

$\Rightarrow y = \dfrac{m$1y2 + m2y1}{m1 + m2} \\[1em] \Rightarrow y = \dfrac{5 \times 2 + 3 \times 7}{5 + 3} \\[1em] \Rightarrow y = \dfrac{10 + 21}{8} \\[1em] \Rightarrow y = \dfrac{31}{8}.

Hence, abscissa of L = 0, m₁ : m₂ = 5 : 3 and co-ordinates of L = $\Big(0, \dfrac{31}{8}\Big)$.