For various a and b, investigate

At first, I will investigate x=cos t and y=sin t

Let's consider when a=b, i.e., x=a cos(t), y=a sin(t)

Then cos(t)=x/a, sin(t)=y/a

so cos^2 (t)+ sin^2(t)= (x/a)^2+(y/a)^2=1

Simplifing this we get, x^2+y^2=a^2

This circle centered origin has radius a ,which a is variable

Now examine the curve when a<b

I let a=1 fixed and b variable

Above movie, This graph is an ellipse centered at origin with vertices at (1,0),(-1,0) and (0,n),(0,-n)

x=cos(t) y=n*sin(t) , By using the fact that cos^2(t) +sin^2(t) =1, we have an equation x^2 + y^2/n^2 = 1

Let's investigate the curve when a>b.

I assume b=1 and a be variable.

By varying a from 0 to 5, the vertices along the X axis vary with respect to n.

This graph is also an ellipse centered at origin with vertices at (0,1),(0,-1) and (n,0),(-n,0)

x=n*cos(t) y= sin(t) , By using the fact that cos^2(t) +sin^2(t) =1,

we get cos(t)=x/n and sin(t)=y so the equation would be x^2/n^2 + y^2 = 1

In conclusion,

I. If a=b then the parametric curve would be a circle with radius a or b.

II. If a is different from b, then the parametric curve would be an ellipse