Let \frac{x-a}{x-b} = m and \frac{a}{b} =n
So we get m + \frac{1}{m} = n + \frac{1}{n}
Or m^2n + n = n^2m + m
Or m^2n - n^2m = m - n
Or mn(m -n ) = m - n
Or m-n = 0 Or mn = 1
case 1
m = n gives \frac{x-a}{x-b} = \frac{b}{a}
Or b(x-a) = x(a-b) or bx-ab = ax-ab or bx = ax or x(a-b)=0 or x = 0
Case 2)
mn =1 gives
\frac{x-a}{x-b}*\frac{a}{b}=1
or a(x-a)=b(x-b)
Or ax-a^2=bx-b^2
Or ax-bx = a^2-b^2
Or x(a-b) = a2-b^2
Or x = a + b
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