Find the ratio in which the line segment joining A(1, -5) and B(-4, 5) is divided by the x-axis. Also find the coordinates of the point of division.

Let point on x-axis be (x, 0) as y-coordinate = 0 on x-axis.

Let ratio be k : 1.

By section-formula,

(x, y) = $\Big(\dfrac{m$1x2 + m2x1}{m1 + m2}, \dfrac{m1y2 + m2y1}{m1 + m2}\Big)

Substituting for y-coordinate we get,

$\Rightarrow 0 = \dfrac{k \times 5 + 1 \times -5}{k + 1} \\[1em] \Rightarrow 0 = \dfrac{5k - 5}{k + 1} \\[1em] \Rightarrow 5k - 5 = 0 \\[1em] \Rightarrow 5k = 5 \\[1em] \Rightarrow k = \dfrac{5}{5} = 1.$

k : 1 = 1 : 1.

Substituting value for x-coordinate, we get :

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

Point = (x, 0) = $\Big(-\dfrac{3}{2}, 0\Big)$.

Hence, ratio = 1 : 1 and point of division = $\Big(-\dfrac{3}{2}, 0\Big)$.