Prove that the locus of feet of perpendiculars from foci upon any tangent to an ellipse is an auxiliary circle

We know that equation of any tangent to ellipse is y=mx+(a^2m^2+b^2)^1/2 ..........(1)Any point F (h,k) lies on tangent k-mh=(a^2m^2+b^2)^1/2 .............(2)Focus S (ae,0) slope of line SF = (k-0)/h-aeSF is perpendicular to line(1) m1m2=-1 k-0/h-ae*m =-1 》m=ae-h/k 》km+h=ae... (4)Square individually 3 and 4 equation and addK^2+m^2+k^2m^2+h^2=a^2m^2+b^2+a^2e^2 》k^2 (1+m^2)+h^2 (1+m^2)=a^2 (1+m^2)》h^2+k^2=a^2》therefore locus is x^2+y^2=a^2Which is equation of auxiliary circle