The line segment joining A (2,3) and B(-3,5) is extended through each end by a length equal to its original length. Find the coordinates of the new ends.

Let P and Q be the required new ends

Coordinates of P

Let AP = k

∴ AB = AP = k

and PB = AP + AB = k + k = 2k

∴ P divides AB externally in the ratio 1:2

Using the section formula for the external division, i.e.

…(i)

Here, m₁ = 1, m₂ = 2

(x₁, y₁) = (2, 3) and (x₂, y₂) = (-3, 5)

Putting the above values in the above formula, we get

⇒ x = 7, y = 1

∴Coordinates of P are (7, 1)

Coordinates of Q.

Q divides AB externally in the ratio 2:1

Again, Using the section formula for the external division, i.e.

…(i)

Here, m₁ = 2, m₂ = 1

(x₁, y₁) = (2, 3) and (x₂, y₂) = (-3, 5)

Putting the above values in the above formula, we get

∴Coordinates of Q are (-8, 7)