Question

In what ratio is the line joining \(\mathrm{P}(5, 3)\) and \(\mathrm{Q}(-5, 3)\) divided by the x-axis? Also find the coordinates of the point of intersection.

Answer 1

Both points have the same y-coordinate ($$y = 3$$), so the line joining them is the horizontal line $$y = 3$$, which runs parallel to the x-axis.

Let us check this with the section formula. Suppose the x-axis cuts this line in the ratio $$k : 1$$ at a point $$(x, 0)$$:

$$\frac{k \times 3 + 1 \times 3}{k + 1} = 0$$

$$\Rightarrow 3k + 3 = 0$$

$$\Rightarrow k = -1$$

A ratio of $$-1 : 1$$ means external division in the ratio $$1 : 1$$ – in other words, the joining line $$y = 3$$ is parallel to the x-axis and has no point of intersection with the x-axis.

For comparison: the y-axis cuts the segment $$PQ$$ in the ratio $$1 : 1$$ at the point $$(0, 3)$$.