2024/001) Given $\frac{x-a}{x-b} + \frac{x-b}{x-a} = \frac{a}{b} + \frac{b}{a}$

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 $x = a + b$