Question

solve by factorisation method x-a/x-b+x-b/x-a=a/b+b/a
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Answer 1

We have,

$:\implies\sf \dfrac{x - a}{x - b} + \dfrac{x - b}{x - a} = \dfrac{a}{b} + \dfrac{b}{a}$

$:\implies\sf \dfrac{(x - a)(x - a) + (x - b)(x - b)}{(x - b)(x - a)} = \dfrac{ a^2 + b^2}{ab}$

$:\implies\sf \dfrac{(x - a)^2 + (x - b)^2}{x^2 - ax - bx + ab} = \dfrac{ a^2 + b^2}{ab}$

$:\implies\sf 2abx^2 - 2abx(a + b) + ab(a^2 + b^2)$

$:\implies\sf (a^2 + b^2)x^2 - (a^2 + b^2)(a + b)x + ab(a^2 + b^2)$

$:\implies\sf (a^2 + b^2 - 2ab)x^2 - (a + b)(a^2 + b^2 -2ab)x = 0$

$:\implies\sf (a - b)^2 x^2 - (a + b)(a - b)^2 x = 0$

$:\implies\sf (a - b)^2 x^2 - (x - (a + b)) = 0$

$:\implies\sf x(x - (a + b)) = 0$

$:\implies\sf x = \bf{0} \; or \; x - (a + b) = 0 \implies x = \bf{a + b}$

$\rule{200}{2}$