Prove that the origin O divides the line segment joining the points, A(1, –3) and B(–3, 9) in the ratio 1 : 3 internally. Find the coordinates of the points dividing externally.

Internal divison formula:

Coordinates of a point P(x,y) dividing the line segment joining A (x₁, y₁) and B (x₂, y₂) in the ratio m:n internally are

Let O divide the line segment joining the given points in the ratio λ:1. Then by internal division formula,

3λ = 1

Hence proved that, the origin O divides the line segment joining the points, A(1, –3) and B(–3, 9) in the ratio 1 : 3 internally.

External divison formula:

Coordinates of a point P(x,y) dividing the line segment joining A (x₁, y₁) and B (x₂, y₂) in the ratio m:n externally are

Using external division formula,

x =

y =

P (3,–9) is the required point.