Question

(x―a)÷(x―b)+(x―b)÷(x―a)=(a÷b)+((b÷a)​​

Answer 1

$\frac{ {(x - a)}^{2} + {(x - b)}^{2} }{(x - a)(x - b) } = \frac{ {a}^{2} + {b}^{2} }{ab} \\ ab( {x}^{2} + {a}^{2} - 2ax + {x}^{2} + {b}^{2} - 2bx = ( {a}^{2} + {b}^{2} )(x - b)(x - a) \\ ab(2 {x}^{2} + {a}^{2} + {b}^{2} - 2x(a + b)) = ({a}^{2} + {b}^{2} )( {x}^{2} - bx - ax + ab) \\ 2 {x}^{2} ab + ( {a}^{2} + {b}^{2} )ab - 2x(a + b)ab = {x}^{2} ( {a}^{2} + {b}^{2} ) - x(a + b)( {a}^{2} + {b}^{2} ) + ab( {a}^{2} + {b}^{2} ) \\ {x( {a}^{2} } + {b}^{2} ) - 2ab {x}^{2} - x(a + b)( {a}^{2} + {b}^{2} ) + 2x(a + b)ab = 0 \\ {x}^{2} ( {a}^{2} + {b}^{2} - 2ab) - x(a + b)( {a}^{2} + {b}^{2} - 2ab) = 0 \\ x {(a - b)}^{2} (x - (a + b)) = 0 \\ x = 0 \: and \: x -(a + b) = 0 \: or \: x = (a + b)$