Question

If x = h + a cos θ, y = k + a sin θ, prove that: (x − h)² + (y − k)² = a²

Answer 1

x = h + a cos θ

⇒ x − h = a cos θ

y = k + a sin θ

⇒ y − k = a sin θ

Square both rearranged equations:

(x − h)² = (a cos θ)²

(x − h)² = a² cos² θ    .....(1)

(y − k)² = (a sin θ)²

(y − k)² = a² sin² θ    .....(2)

Add Equations 1 and 2:

(x − h)² + (y − k)² = a² cos² θ + a² sin² θ

(x − h)² + (y − k)² = a² (cos² θ + sin² θ)

Use the fundamental identity sin² θ + cos² θ = 1

(x − h)² + (y − k)² = a² (1)

(x − h)² + (y − k)² = a²

Hence Proved.