Question
Simplify$:\frac{\sqrt{x^2+y^2}-y}{x-\sqrt{x^2-y^2}} \div \frac{\sqrt{x^2-y^2}+x}{\sqrt{x^2+y^2}+y}$
✓
Answer
$\frac{\sqrt{x^2+y^2}-y}{x-\sqrt{x^2-y^2}} \div \frac{\sqrt{x^2-y^2}+x}{\sqrt{x^2+y^2}+y}$
$\Rightarrow \frac{\sqrt{x^2+y^2}-y}{x-\sqrt{x^2-y^2}} \times \frac{\sqrt{x^2+y^2}+y}{\sqrt{x^2-y^2}+x}$
$\Rightarrow \frac{\left(\sqrt{x^2+y^2}-y\right)\left(\sqrt{x^2+y^2}+y\right)}{\left(x-\sqrt{x^2-y^2}\right)\left(x+\sqrt{x^2-y^2}\right)}$
$\Rightarrow \frac{\left(\sqrt{x^2+y^2}\right)^2-y^2}{x^2-\left(\sqrt{x^2-y^2}\right)^2}$
$\Rightarrow \frac{x^2+y^2-y^2}{x^2-\not x^2+y^2}$
$\Rightarrow \frac{x^2}{y^2}$
$\Rightarrow \frac{\sqrt{x^2+y^2}-y}{x-\sqrt{x^2-y^2}} \times \frac{\sqrt{x^2+y^2}+y}{\sqrt{x^2-y^2}+x}$
$\Rightarrow \frac{\left(\sqrt{x^2+y^2}-y\right)\left(\sqrt{x^2+y^2}+y\right)}{\left(x-\sqrt{x^2-y^2}\right)\left(x+\sqrt{x^2-y^2}\right)}$
$\Rightarrow \frac{\left(\sqrt{x^2+y^2}\right)^2-y^2}{x^2-\left(\sqrt{x^2-y^2}\right)^2}$
$\Rightarrow \frac{x^2+y^2-y^2}{x^2-\not x^2+y^2}$
$\Rightarrow \frac{x^2}{y^2}$
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