# Parametric Equations of a Circle

## Parametric Equations of a Circle

If P (x, y) is a point on the circle with center C (h, k) and radius r, then x = h + r sin θ, y = k = r cos θ where 0 ≤ θ < 2π.

Proof: Let θ be the angle made by the line $$\overleftrightarrow{CP}$$ with x – axis in the positive direction let D, M be the projections of C, P on x – axis respectively. Let Q be the projection of C on PM.

Now ∠PCQ = θ

cos θ = CQ/ CP,

sin θ = PQ/ CP

Therefore CQ = CP Cosθ

= r Cosθ

PQ = CP sinθ

= r sinθ

x = OM = OD + DM

=OD + CQ

= h + r cosθ

Y = PM = PQ + QM

= PQ + CD

= k + r sinθ

A point on the circle x² + y² = r² is taken in the form (r cosθ, r sinθ). The point (r cosθ, r sinθ). is simply denoted as point θ.

Example 1: The parametric equation of the circle (x-3) ² + (y-6) ² = 8².

Solution: Given that

(x-3) ² + (y-6) ² = 8²

Above equation h = 3, k = 6 and r = 8

The parametric equations are

x = h + r cosθ

y = k + r sinθ

x = 3 + 8 cosθ

y = 6 + 8 sinθ

where 0 ≤ θ < 2π

Example 2: Find the parametric equation of the circle 4(x² + y²) = 9.

Solution: Given circle equation is 4(x² + y²) = 9

(x² + y²) = 9/4

x² + y² = (3/2)² … (1)

center (h, k) = (0, 0)

radius (r) = 3/2

the parametric equation are

x = h + r cosθ

y = k + r sinθ

x = 0 + 3/2 cosθ,

y = 0 + 3/2 sinθ. where 0 ≤ θ < 2π.