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Mathematics

Simplify :

x2+y2−yx−x2−y2÷x2−y2+xx2+y2+y\dfrac{\sqrt{x^2 + y^2} - y}{x - \sqrt{x^2 - y^2}} ÷ \dfrac{\sqrt{x^2 - y^2} + x}{\sqrt{x^2 + y^2} + y}

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Answer

Solving,

⇒x2+y2−yx−x2−y2÷x2−y2+xx2+y2+y⇒x2+y2−yx−x2−y2×x2+y2+yx2−y2+x⇒(x2+y2)2−y2x2−(x2−y2)2⇒x2+y2−y2x2−(x2−y2)⇒x2y2.\Rightarrow \dfrac{\sqrt{x^2 + y^2} - y}{x - \sqrt{x^2 - y^2}} ÷ \dfrac{\sqrt{x^2 - y^2} + x}{\sqrt{x^2 + y^2} + y} \\[1em] \Rightarrow \dfrac{\sqrt{x^2 + y^2} - y}{x - \sqrt{x^2 - y^2}} \times \dfrac{\sqrt{x^2 + y^2} + y}{\sqrt{x^2 - y^2} + x} \\[1em] \Rightarrow \dfrac{(\sqrt{x^2 + y^2})^2 - y^2}{x^2 - (\sqrt{x^2 - y^2})^2 } \\[1em] \Rightarrow \dfrac{x^2 + y^2 - y^2}{x^2 - (x^2 - y^2)} \\[1em] \Rightarrow \dfrac{x^2}{y^2}.

Hence, x2+y2−yx−x2−y2÷x2−y2+xx2+y2+y=x2y2\dfrac{\sqrt{x^2 + y^2} - y}{x - \sqrt{x^2 - y^2}} ÷ \dfrac{\sqrt{x^2 - y^2} + x}{\sqrt{x^2 + y^2} + y} = \dfrac{x^2}{y^2}.

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