The line segment joining A(4, 7) and B(-6, -2) is intercepted by the y-axis at the point K. Write down the abscissa of the point K. Hence, find the ratio in which K divides AB. Also, find the co-ordinates of the point K.

Since, point K lies on y-axis. Its co-ordinates be (0, y).

Let ratio in which K divides AB be m₁ : m₂.

By section-formula,

$x = \dfrac{m$1x2 + m2x1}{m1 + m2} \\[1em] \Rightarrow 0 = \dfrac{m1 \times -6 + m2 \times 4}{m1 + m2} \\[1em] \Rightarrow 0 = -6m1 + 4m2 \\[1em] \Rightarrow 6m1 = 4m2 \\[1em] \Rightarrow \dfrac{m1}{m2} = \dfrac{4}{6} = \dfrac{2}{3}.

m₁ : m₂ = 2 : 3.

Substituting value for y co-ordinate,

$\Rightarrow y = \dfrac{m$1y2 + m2y1}{m1 + m2} \\[1em] = \dfrac{2 \times -2 + 3 \times 7}{2 + 3} \\[1em] = \dfrac{-4 + 21}{5} \\[1em] = \dfrac{17}{5}.

Hence, abscissa of K = 0, ratio in which K divides AB = 2 : 3 and K = $\Big(0, \dfrac{17}{5}\Big).$