Question

Find the ratio in which the line segment joining the points (−2, 5) and (3, 7) is divided by the Y-axis.

Answer 1

If P divides AB in the ratio m : n (AP : PB = m : n) with A(x₁, y₁) and B(x₂, y₂),

Then, P = `((mx_2 + nx_1) / (m + n)), ((my_2 + ny_1) / (m + n))`

Here let, A = (−2, 5) = (x₁, y₁),

B = (3, 7) = (x₂, y₂)

And let, AP : PB = k : 1 (so m = k, n = 1)

Use the x‑coordinate because the Y‑axis has x = 0,

Put x = 0 in the section formula:

0 = `(kx_2 + 1·x1) / (k + 1)` 

0 = `(k3 + (−2)) / (k + 1)`

0 = `(3k − 2) / (k + 1)`

3k − 2 = 0

∴ k = `2 / 3`

The ratio and the intersection point:

AP : PB = k : 1

= `(2 / 3) : 1`

= 2 : 3

Hence, the Y‑axis divides the segment joining (−2, 5) and (3, 7) in the ratio 2 : 3 (AP : PB).