For various values of a and b investigate the parametric equations x=acos(t), y=bsin(t) for t between 0 and 2pi.


First, observe that, for the given domain, both sine and cosine will run through a full period. Thus, for any graph that we draw we will see all there is to see by letting t run from 0 to 2pi.

Note that when I say minor/major axis here you are actually getting half the full length.

If a=1, b=1 then we know that this will be a circle (since x^2+y^2=1). Let's explore some other values:

a=1 b=2 gives a vertical ellipse whose major axis has (center to edge) length 2 (b=2) and minor axis is length 1 (a=1)

Now, let's sketch for a=2, b=1:



This mathematics should convince you that we get an ellipse for all nonzero values of a and b.