Question : The simplified value of $\small \left (1-\frac{2xy}{x^{2}+y^{2}}\right )\div\left (\frac{x^{3}-y^{3}}{x-y}-3xy\right)$ is:
Option 1: $\frac{1}{x^{2}-y^{3}}$
Option 2: $\frac{1}{x^{2}+y^{2}}$
Option 3: $\frac{1}{x-y}$
Option 4: $\frac{1}{x+y}$
Correct Answer: $\frac{1}{x^{2}+y^{2}}$
Solution :
Given:
$(1–\frac{2xy}{x^{2}+y^{2}})\div( \frac{x^{3}–y^{3}}{x–y}–3xy)$
= $\frac{(x^{2}+y^{2}–2xy}{x^{2}+y^{2}})\div( \frac{(x–y)(x^{2}+xy+y^{2})}{x–y}–3xy)$
= $\frac{(x–y)^{2}}{x^{2}+y^{2}}\div(x^{2}+xy+y^{2}–3xy)$
= $\frac{(x–y)^{2}}{x^{2}+y^{2}}\div(x^{2}+y^{2}–2xy)$
= $\frac{(x–y)^{2}}{x^{2}+y^{2}}\div(x–y)^{2}$
= $\frac{1}{x^{2}+y^{2}}$
Hence, the correct answer is $\frac{1}{x^{2}+y^{2}}$.
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