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

We have,

{tex}\frac{{x - a}}{{x - b}} + \frac{{x - b}}{{x - a}} = \frac{a}{b} + \frac{b}{a}{/tex}
{tex}\Rightarrow \frac{{(x - a)(x - a) + (x - b)(x - b)}}{{(x - b)(x - a)}} = \frac{{{a^2} + {b^2}}}{{ab}}{/tex}
{tex} \Rightarrow \frac{{{x^2} + {a^2} - 2ax + {x^2} + {b^2} - 2bx}}{{{x^2} - bx - ax + ab}}{/tex} {tex} = \frac{{{a^2} + {b^2}}}{{ab}}{/tex}
{tex}\Rightarrow \frac{{2{x^2} - 2ax - 2bx + {a^2} + {b^2}}}{{{x^2} - bx - ax + ab}} = \frac{{{a^2} + {b^2}}}{{ab}}{/tex}
{tex}\Rightarrow{/tex}{tex} (2x^2 - 2ax - 2bx + a^2 + b^2)ab = (a^2 + b^2)(x^2 - bx - ax + ab){/tex}
{tex}\Rightarrow{/tex} {tex}2abx^2 - 2a^2bx - 2ab^2x + a^3b + ab^3 = a^2x^2 - a^2bx -a^3x + a^3b + b^2x^2 - b^3x - ab^2x + ab^3{/tex}
{tex}\Rightarrow{/tex} {tex}2abx^2 - a^2x^2 - a^2bx - ab^2x + a^3x + b^3x - b^2x^2 = 0{/tex}
{tex}\Rightarrow{/tex} {tex}(2ab - a^2 - b^2)x^2 + (-a^2b - ab^2 + a^3 + b^3)x = 0{/tex}
{tex}\Rightarrow{/tex} x[(2ab - a² - b²)x + (a³ + b³ - a²b - ab²)] = 0
{tex}\Rightarrow{/tex} x = 0 or (2ab - a² - b²)x + a³ + b³ -a²b - ab² = 0
Now,
(2ab - a² - b²)x + a³ + b³ - a²b - ab² = 0
{tex}\Rightarrow{/tex} (2ab - a² - b²)x = a²b + ab² - a³ - b³
{tex}\Rightarrow{/tex} -(a² - b² - 2ab)x = a²b - b³ + ab² - a³
{tex}\Rightarrow{/tex} -(a - b)²x = b(a² - b²) + a(b² - a²)
{tex}\Rightarrow{/tex} (a - b)²x = -b(a² - b²) - a(b² - a²)
{tex} \Rightarrow x = \frac{{ - b({a^2} - {b^2}) + a({a^2} - {b^2})}}{{{{(a - b)}^2}}}{/tex}
{tex}\Rightarrow x = \frac{{\left( {{a^2} - {b^2}} \right)(a - b)}}{{{{(a - b)}^2}}}{/tex}
{tex} \Rightarrow x = \frac{{{a^2} - {b^2}}}{{a - b}} = \frac{{(a - b)(a + b)}}{{(a - b)}}{/tex}
{tex}\Rightarrow{/tex} x = a + b
{tex}\therefore{/tex} x = 0 or x = a + b