<s>[INST] Is this an ellipse? 
Is this parameterisation an ellipse:
  \begin{align}x(t) &= \frac{2 \cos(t)}{1 + a \sin(t)}\\
y(t) &= \frac{2 \sin(t)}{1 + a \sin(t)}\end{align}
   where $a$ is a real positive parameter.

I tried to do it the naive way but couldn't find a definitive answer.
Plotting our curve with the help of Geogebra gives the following very ellipse like graph: 
 
Any help would be appreciated.
 [/INST] $$\begin{align}
x&=\frac {2\cos t}{1+a\sin t}\tag{1}\\
y&=\frac {2\sin t}{1+a\sin t}\tag{2}\\
(2)/(1):\hspace{3cm}\\
\frac yx&=\tan t\tag{3}\\
(1)^2+(2)^2:\hspace{3cm}\\
x^2+y^2&=\frac 4{(1+a\sin t)^2}\\
&=\frac {4(x^2+y^2)}{\big(\sqrt{x^2+y^2}+ay\big)^2}
&&\scriptsize \bigg(\sin t=\frac y{\sqrt{x^2+y^2}}\bigg)\\
(x^2+y^2)\big[\big(\sqrt{x^2+y^2}+ay\big)^2-4\big]&=0\\
\because{x^2+y^2}\neq 0\therefore \qquad
\big(\sqrt{x^2+y^2}+ay\big)^2-4&=0\\
x^2+y^2&=(\pm2-ay)^2\\
\color{red}{x^2+(1-a^2)y^2\pm 4ay-4}&\color{red}{=0}
\end{align}$$
which is an ellipse if $a^2<1$, per criteria outlined here.
</s>