Ok, so if the formula for an Archimedean spiral does not have a straightforward Cartesian form solved for y in terms of x, how does one find the intercept between said spiral and another curve?

In particular I need the first quadrant intercept(s?) between the Archimedean spiral r=x(b+1)/pi and the ellipse arc r=sqrt(1/(1+cos(x)^2)) (which is of course y=sqrt(b^2-b^2x^2) in Cartesian).