Question

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

(6, −4), (5, 5)

Answer 1

Here, let A = (6, −4) = (x₁, y₁) and B = (5, 5) = (x₂, y₂),

Let the X-axis meet AB at P so that AP : PB = k : 1

If a point divides AB in the ratio k : 1 (AP : PB = k : 1),

Then, x = `(kx_2 + x_1) / (k + 1) and y = (ky_2 + y_1) / (k + 1)`

P lies on the X‑axis, so its y‑coordinate is 0.

Apply the y‑formula:

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

0 = k × 5 + (−4)

5k − 4 = 0

∴ k = `4 / 5`

Convert k : 1 to an integer ratio:

AP : PB = k : 1

= `4/5` : 1

= 4 : 5

Hence, the X‑axis divides the segment joining (6, −4) and (5, 5) in the ratio 4 : 5 (AP : PB).