solve for x :-

(x-a)/(x-b) + (x-b)/(x-a) = a/b + b/a

x-a/x-b -a/b + x-b/x-a -b/a =0

bx -ax/b(x-b)+ ax-bx/a(x-a)=0

bx-ax/b(x-b) = bx-ax/a(x-a)

1/bx-b2 = 1/ax-a2

ax-a² = bx-b²

ax-bx=a²-b²

x=a+b

Cheers!