conic sections – Locus of focus of ellipse which touches the x axis

An ellipse touches the x axis. The length of major axis is 2a while the minor axis is given as 2b, What would be the locus of it’s focus? (a>b)

My Approach: If the focus is (x1,y1) and (x2,y2) then $(x1-x2)^2 + (y1 – y2)^2 = 4(a^2 – b^2)$ also $y1y2 = b^2$ since product of perpendicular subtended from the focii on any tangent is $b^2$. These are the properties which seem relevant in this question but I am not able to use them to find out the answer. I also might be missing out on other things.

Any hints/solutions to the problem are appreciated.