Question

If P(x, y) lies on the interval A(x₁, y₁), B(x₂, y₂) such that AP : PB = a : b, with a and b positive, show that

x = (bx₁ ax₂)/(b + a) and y = (by₁ + ay₂)/(b + a).

Answer 1

Let A(x₁, y₁) and B(x₂, y₂) be two points. Assume that P(x, y) is on the interval AB with

AP : PB = a : b = a b : 1. 

So P divides AB in the ratio a b : 1. Thus

x = (x₁ + (a/b)x₂)/(1 + a/b) = (bx₁ + ax₂)/(b + a)

and, similarly

y = (by₁ + ay₂)/(b + a)