Question

Simplify:
root x square + y square minus y upon x minus root x square minus y square divided by root x square minus y square + X upon root x square + y square + y

Answer 1

$\textbf{Given:}$

$\mathsf{\dfrac{\dfrac{\sqrt{x^2+y^2}-y}{x-\sqrt{x^2-y^2}}}{\dfrac{\sqrt{x^2-y^2}+x}{\sqrt{x^2+y^2}+y}}}$

$\textbf{To simplify:}$

$\mathsf{\dfrac{\dfrac{\sqrt{x^2+y^2}-y}{x-\sqrt{x^2-y^2}}}{\dfrac{\sqrt{x^2-y^2}+x}{\sqrt{x^2+y^2}+y}}}$

$\textbf{Solution:}$

$\textsf{Consider,}$

$\mathsf{\dfrac{\dfrac{\sqrt{x^2+y^2}-y}{x-\sqrt{x^2-y^2}}}{\dfrac{\sqrt{x^2-y^2}+x}{\sqrt{x^2+y^2}+y}}}$

$\textsf{This can be written as}$

$\mathsf{=\dfrac{\sqrt{x^2+y^2}-y}{x-\sqrt{x^2-y^2}}{\times}\dfrac{\sqrt{x^2+y^2}+y}{\sqrt{x^2-y^2}+x}}$

$\mathsf{=\dfrac{\sqrt{x^2+y^2}-y}{x-\sqrt{x^2-y^2}}{\times}\dfrac{\sqrt{x^2+y^2}+y}{x+\sqrt{x^2-y^2}}}$

$\textsf{Using the identity,}$

$\boxed{\mathsf{(a-b)(a+b)=a^2-b^2}}$

$\mathsf{=\dfrac{(\sqrt{x^2+y^2})^2-y^2}{x^2-(\sqrt{x^2-y^2})^2}}$

$\mathsf{=\dfrac{x^2+y^2-y^2}{x^2-(x^2-y^2)}}$

$\mathsf{=\dfrac{x^2+y^2-y^2}{x^2-x^2+y^2}}$

$\mathsf{=\dfrac{x^2}{y^2}}$

$\implies\boxed{\mathsf{\dfrac{\dfrac{\sqrt{x^2+y^2}-y}{x-\sqrt{x^2-y^2}}}{\dfrac{\sqrt{x^2-y^2}+x}{\sqrt{x^2+y^2}+y}}=\dfrac{x^2}{y^2}}}$

$\textbf{Find more:}$

Simplify 73*73*73+27*27*27/73*73-73*27+27*27

https://brainly.in/question/2079320