How to prove the parametric equation of an ellipse? The parametric equation of an ellipse is
$$x=a \cos t\\y=b \sin t$$
It can be viewed as $x$ coordinate from circle with radius $a$, $y$ coordinate from circle with radius $b$.

How to prove that it's an ellipse by definition of ellipse (a curve on a plane that surrounds two focal points such that the sum of the distances to the two focal points is constant for every point on the curve) without using trigonometry and standard equation of ellipse?

Why not simply
\begin{equation*}
\left(\frac{x}{a}\right)^{2}+\left(\frac{y}{b}\right)^{2}=\cos ^{2}t+\sin ^{2}t=1
\end{equation*}
