5.3 The circle and the ellipse
The equation of a circle can be formed easily if we recall the Pythagoras' theorem. again. A circle with centre at point $C=(c_1,c_2)$ and radius $r>0$ is the set of all points $(x,y)$ whose distance from $C$ is equal to $r$. Hence
\begin{equation*}
(x-c_1)^2 + (y-c_2)^2 = r^2
\end{equation*}
This situation is shown in Figure 5.3.
The equation of an ellipse is given by
\begin{equation*}
\frac{(x-c_1)^2}{a^2} + \frac{(y-c_2)^2}{b^2} = 1,
\end{equation*}
where $a$ and $b$ are positive parameters and $A = (c_1,c_2)$ is the centre of the ellipse. The parameters $a$ and $b$ define the length of the semi-major axis and the semi-minor axis. If $a=b$ then we get a circle. A typical ellipse is depicted in Figure 5.4.
