In what ratio does the x-axis divide the line segment joining the points (–4, –6) and (–1, 7)? Find the co-ordinates of the point of division.

Let A(-4, -6) and B(-1, 7) be the points.

Let P(x, 0) be the point which divide the line segment in some ratio say, k:1.

We need to find this ratio k:1.

Here, x₁ =-4 and y₁ = -6

x₂ = -1 and y₂ = 7

m₁ = k and m₂ = 1

x = x and y = 0

We know the section formula,

⇒

⇒

⇒ 7k – 6 = 0

⇒ 7k = 6

⇒ k = 6/7

Thus, the ratio is 6/7:1.

Simplifying it by multiplying 7 on both sides,

Hence, the ratio is 6:7.

Also, we need to find coordinates of the point of division, i.e., P(x, 0).

⇒ We need to find x.

We have got

x₁ =-4 and y₁ = -6

x₂ = -1 and y₂ = 7

m₁ = 6 and m₂ = 7.

Using section formula for x,

⇒

⇒

Hence, the coordinate of the point dividing the line segment in 6:7 ratio is (-34/13,0).