# Equation of Circle/Parametric

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## Theorem

The equation of a circle with radius $R$ and center $\left({a, b}\right)$ expressed as a parametric equation

- $x = a + R \cos t, \ y = b + R \sin t$

## Proof

Let the point $\left({x, y}\right)$ satisfy the equations:

- $x = a + R \cos t$
- $y = b + R \sin t$

By the Distance Formula, the distance between $\left({x, y}\right)$ and $\left({a, b}\right)$ is:

- $\sqrt{\left({\left({a + R \cos t}\right) - a}\right)^2 + \left({\left({b + R \sin t}\right) - b}\right)^2}$

This simplifies to:

- $\sqrt{R^2 \cos^2 t + R^2 \sin^2 t} = R \sqrt{\cos^2 t + \sin^2 t}$

Then by Sum of Squares of Sine and Cosine, this distance equals $R$.

Therefore the distance between points satisfying the equation and the center is constant and equal to the radius.

$\blacksquare$