Question

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

(5, 4), (1, −3)

Answer 1

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

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

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

Then P has coordinates:

P = `(((k xx 1) + 5)/(k + 1)), (((k xx (−3)) + 4)/(k + 1))`

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

So set the y-coordinate equal to 0:

`(k(−3) + 4) / (k + 1) = 0`

−3k + 4 = 0

∴ k = `4 / 3`

Convert k : 1 to an integer ratio:

k : 1 = `(4/3)` : 1

∴ k = 4 : 3

Hence, the x-axis divides the segment joining (5, 4) and (1, −3) in the ratio 4 : 3.