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Question

Solve 
$$xy[(x+y)(\dfrac{1}{x} +\dfrac{1}{y})- y ]= (x-y)^2$$


Solution

$$xy \left[(x + y) \left(\dfrac{1}{x} + \dfrac{1}{y}\right) - y \right] = (x - y)^2$$
$$xy \left[(x + y) \dfrac{(x + y)}{xy} - y\right] = (x - y)^2$$
$$xy \left[\dfrac{(x + y)^2 - xy^2}{xy} \right] = ( x - y)^2$$
$$x^2 + y^2 + 2xy - xy^2 = x^2 + y^2 - 2xy$$
$$4 xy - xy^2 = 0$$
$$xy (4 - y) = 0$$
$$xy = 0 , \, y = 4$$

Mathematics

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