Parametric Equation

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 a circle by relating the coordinates with a ‘parameter’.

Parametric Equation for the Standard Circle

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’ll get:

x = rcosθ

y = rsinθ

And that’s it! We got what’s called the parametric equation of the circle. Here, θ is the parameter, which represents the angle made by OP with the X-axis.

In other words, for all values of θ, the point (rcosθ, rsinθ) lies on the circle x² + y² = r². 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.

Parametric Equation for the General Circle

Consider the general equation of the circle:

x² + y² + 2gx + 2fy + c = 0

This can be written as:

(x + g)² + (y + f)² = r² (where r² = g² + f² – 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’ll get

x + g = rcosθ

y + f = rsinθ

This gives us

x = –g + rcosθ

y = -f + rsinθ

And, this is the parametric equation of the circle x² + y² + 2gx + 2fy + c = 0.

Here’s a simulation that demonstrates the parametric equation of a circle.

You can drag the slider to change the value of θ, and observe the point (rcosθ, rsinθ). Does it always lie on the circle?

Lesson Summary

1. The parametric equation of the circle x² + y² = r² is x = rcosθ, y = rsinθ.
2. The parametric equation of the circle x² + y² + 2gx + 2fy + c = 0 is x = -g + rcosθ, y = -f + rsinθ.

Here, 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!