In what ratio is the line joining A(0, 3) and B (4, -1) divided by the x-axis ?

Write the co-ordinates of the point where AB intersects the x-axis.

Let AB intersect x-axis at P. So, co-ordinates of P = (x, 0).

Let ratio be m₁ : m₂.

By section formula,

$y = \dfrac{m$1y2 + m2y1}{m1 + m2} \\[1em] \Rightarrow 0 = \dfrac{m1 \times -1 + m2 \times 3}{m1 + m2} \\[1em] \Rightarrow 0 = -m1 + 3m2 \\[1em] \Rightarrow m1 = 3m2 \\[1em] \Rightarrow \dfrac{m1}{m2} = \dfrac{3}{1}.

m₁ : m₂ = 3 : 1.

$x = \dfrac{m$1x2 + m2x1}{m1 + m2} \\[1em] = \dfrac{3 \times 4 + 1 \times 0}{3 + 1} \\[1em] = \dfrac{12 + 0}{4} \\[1em] = \dfrac{12}{4} = 3. \\[1em]

P = (x, 0) = (3, 0).

Hence, co-ordinates of P = (3, 0) and ratio = 3 : 1.