This lesson will cover the parametric equation of a circle.

Just like the parametric equation of a line, this form will help us to find the coordinates of any point on the circle by relating the coordinates with a ‘parameter’.

Consider the following circle, whose center is at O(0, 0) and radius equals r.

Let P(x, y) be any point on the circle such the OP makes an angle θ with the X axis. Using trigonometry, we have x = rcosθ and y = rsinθ.

And that’s it! We have what’s called the parametric equation of the circle: **x = rcosθ, y = rsinθ** (where θ is a parameter).

In other words, for all values of θ, the point (rcosθ, rsinθ) lies on the circle. Or, any point on the circle is (rcosθ, rsinθ), where θ is a parameter.

We can also obtain the parametric equation of the circle whose center does not lie at the origin. Consider the general equation of the circle: x^{2} + y^{2} + 2gx + 2fy + c = 0, which can be written as (x + g)^{2} + (y + f)^{2} = r^{2}, where r^{2} = g^{2} + f^{2} – c.

Again, let P(x, y) be any point on the circle such that CP makes and angle θ with the X-axis.

In this case we have, x + g = rcosθ, and y + f = rsinθ, or **x = **–**g + rcosθ**, **y = -f + rsinθ**, which is the parametric equation of the circle.

To summarize:

(1) The parametric equation of the circle x^{2} + y^{2} = r^{2} is **x = rcosθ, y = rsinθ**

(2) The parametric equation of the circle x^{2} + y^{2} + 2gx + 2fy + c = 0 is **x = **–**g + rcosθ**, **y = -f + rsinθ**

where θ is a parameter, which represents the angle made by the line, joining the point (x, y) with the center, with the X-axis.

That’s it for this lesson. See you in the next one!