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Monday, January 1, 2024

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



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