Ellipse on complex plane

Do the variable substitution $x=tan cfrac{t}{2},,text{where }tin[-pi,pi]$. This equation has a unique solution respect $t$. Further simplify and get $z=cfrac{b}{2}+cfrac{b}{2}cos t +cfrac{a}{2}sin t $ The latter is obviously the equation of an ellipse.

$z=frac{ax+b}{x^2+1}$

where x is just a parameter, so substitute it as t

let a and b be $a_1+ia_2$ and $b_1+ib_2$,

Clearly, Re(z)= $frac{a_1t+b_1}{t^2+1}$ and Im(z)= $frac{a_2t+b_2}{t^2+1}$

Re(z) is plotted on x axis and Im(z) is plotted on y, so simply remove t from both equations simultaneously and make it in cartesian form. Following that, check the eccentricity (there's a general formula as well) and it indeed comes out less than or equal to 1.

https://www.desmos.com/calculator/xjyxtjfyio

Another reason for it being elliptical is that, the function z is bounded in real domain i.e. $(ax+b)/(x^2+1)$ never goes above a finite value.