Question

Find the ratio in which the line segment joining the following points is divided by the X-axis:

(3, −2), (2, 3)

Answer 1

If A(x₁, y₁) and B(x₂, y₂) are joined and a point P divides AB in the ratio k : 1 (AP : PB = k : 1),

Then, P = `((kx_2 + x_1) / (k + 1)), ((ky_2 + y_1) / (k + 1))`

Here,

A = (3, −2) so, x₁ = 3, y₁ = −2;

B = (2, 3) so x₂ = 2, y₂ = 3

Let the x-axis meet AB at P and let AP : PB = k : 1.

Since P lies on the x-axis, its y-coordinate is 0.

Apply the y-coordinate part of the section formula:

0 = `(ky_2 + y_1) / (k + 1)`

0 = `(k3 + (−2)) / (k + 1)`

0 = `(3k − 2)/(k + 1)`

3k − 2 = 0

∴ k = `2/3`

Convert k : 1 = `(2/3)` : 1 to integer ratio by multiplying by 3, so 2 : 3

Therefore, the x-axis divides the segment internally in the ratio 2 : 3 (AP : PB = 2 : 3).