Parametric Equations

Any point on the circle (x−h)² + (y−k)² = r² in terms of the parameter θ ∈ [0°, 360°). For unit circle centred at origin: (cos θ, sin θ).

Consider the circle $(x - h)^2 + (y - k)^2 = r^2$. Any point $P$ on this circle lies at distance $r$ from $(h, k)$. Let $\theta$ be the angle the radius $CP$ makes with the positive $x$-axis.

By the definition of cosine and sine:

$$x - h = r\cos\theta, \quad y - k = r\sin\theta$$

Therefore:

$$x = h + r\cos\theta, \quad y = k + r\sin\theta, \quad \theta \in [0°, 360°)$$

Verification: Substituting back,

$$(r\cos\theta)^2 + (r\sin\theta)^2 = r^2(\cos^2\theta + \sin^2\theta) = r^2 \checkmark$$

For the circle $x^2 + y^2 = a^2$, the parametric point is simply $(\cos\theta, \sin\theta)$ scaled by $a$, written as $(a\cos\theta, a\sin\theta)$.

The parametric form is indispensable in chord and tangent problems where the point of contact is unknown and is better handled as an angle than as $(x_1, y_1)$.