Example I Suppose the parametric equations $\left\{\begin{array}{l}x=\frac{a t}{1+t^{2}}, \\ y=\frac{1-t^{2}}{1+t^{2}},\end{array}\right.$ where $a \in \mathbf{R}^{\prime}$, what type of curve does it represent?
Solving for $x=\frac{a t}{1+t^{2}}$ yields $\frac{2 x}{a}=\frac{2 t}{1+t^{2}}$. Squaring and adding this to $y=\frac{1-t^{2}}{1+t^{2}}$, we get
$$
\left(\frac{2 x}{a}\right)^{2}+y^{2}=1
$$

If $a=2$, equation $(*)$ represents a unit circle;
If $a \neq 2(a>0)$, equation $(*)$ represents an ellipse centered at the origin.