If the line segment joining the points A(a, b) and B(c, d) subtends a right angle at the origin, show that ac + bd = 0.

Given that the line segment joining the points A(a, b) and B(c, d) subtends a right angle at the origin O(0, 0)

So, AOB is a right angled triangle with right angle at O.

We got OA² + OB² = AB² [By Pythagoras Theorem]

We know that the distance between the points (x₁, y₁) and (x₂, y₂) is .

⇒ OA² + OB² = AB²

⇒ (0 - a)² + (0 - b)² + (0 - c)² + (0 - d)² = (a - c)² + (b - d)²

⇒ a² + b² + c² + d² = a² + c² - 2ac + b² + d² - 2bd

⇒ 2ac + 2bd = 0

⇒ ac + bd = 0