TITLE: Is this an ellipse?
QUESTION [30 upvotes]: 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.

REPLY [3 votes]: Renaming $t$ as $\theta$, you have a polar equation:
$$\rho=\frac2{1+a\sin\theta}.$$
This is the polar form of the equation of a conic with a focus at the origin. https://en.wikipedia.org/wiki/Ellipse#Polar_form_relative_to_focus
If $|a|<1$, the denominator has no root hence the curve is bounded.

By rewriting
$$\rho+ay=2$$ you establish
$$x^2+y^2=(2-ay)^2.$$
This conic is an ellipse when the discriminant of the quadratic terms $x^2+(1-a^2)y^2$ is negative, i.e. when $|a|<1$.