In what ratio is the line joining the points (4, 2) and (3, -5) divided by the x-axis? Also find the coordinates of the point of division.

Let the point P which is on the x-axis, divide the line segment joining the points A(4, 2) and B(3, -5) in the ratio of m : n. Let the coordinates of P be (x, 0).

By section formula,

y-coordinate = $\dfrac{my$2 + ny1}{m + n}

$\Rightarrow 0 = \dfrac{m \times (-5) + n \times 2}{m + n} \\[1em] \Rightarrow 0 = -5m + 2n \\[1em] \Rightarrow 5m = 2n \\[1em] \Rightarrow \dfrac{m}{n} = \dfrac{2}{5} \\[1em] \Rightarrow m : n = 2 : 5.$

Putting value of m : n in section-formula for x-coordinate,

x-coordinate = $\dfrac{mx$2 + nx1}{m+ n}

$= \dfrac{2 \times 3 + 5 \times 4}{2 + 5} \\[1em] = \dfrac{6 + 20}{7} \\[1em] = \dfrac{26}{7}.$

Hence, coordinates of P are $(\dfrac{26}{7}, 0)$ and 2 : 5 is the ratio in which the line joining the points (4, 2) and (3, -5) is divided by the x-axis.