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

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

Circle Parametric Equation

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: x2 + y2 + 2gx + 2fy + c = 0, which 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.

circle-parametric-equation-2

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.

 

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

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!

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