Question

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If $\mathrm{x=\dfrac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}}$
&
$\mathrm{y=\dfrac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}}$


then find

 $\sf{\dfrac{x^{2}+xy+y^{2}}{x^{2}-xy+y^{2}}}$

​

Answer 1

Given:-

•$\mathrm{x=\dfrac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}}$

•$\mathrm{y=\dfrac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}}$

To Find:-

•Value of  $\sf{\dfrac{x^{2}+xy+y^{2}}{x^{2}-xy+y^{2}}}$

Solution:-

$\sf Here$

$\leadsto \mathrm{x+y=\dfrac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}+\dfrac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}}$

$\leadsto \mathrm{x+y=\dfrac{(\sqrt{5}-\sqrt{3})^{2}+(\sqrt{5}+\sqrt{3})^{2}}{(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})}}$

$\leadsto \mathrm{x+y=\dfrac{5+3-2\sqrt{5}\sqrt{3}+5+3+2\sqrt{5}\sqrt{3}}{5-3}}$

$\leadsto \mathrm{x+y=\dfrac{8-2\sqrt{15}+8+2\sqrt{15}}{2}}$

$\leadsto \mathrm{x+y=\dfrac{16}{2}}$

$\leadsto \mathrm{x+y=8}$

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$\sf Again$

$\leadsto \mathrm{x-y=\dfrac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}-\dfrac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}}$

$\leadsto \mathrm{x-y=\dfrac{(\sqrt{5}-\sqrt{3})^{2}-(\sqrt{5}+\sqrt{3})^{2}}{(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})}}$

$\leadsto \mathrm{x-y=\dfrac{5+3-2\sqrt{5}\sqrt{3}-(5+3+2\sqrt{5}\sqrt{3})}{5-3}}$

$\leadsto \mathrm{x-y=\dfrac{5+3-2\sqrt{5}\sqrt{3}-5-3-2\sqrt{5}\sqrt{3}}{2}}$

$\leadsto \mathrm{x-y=\dfrac{8-2\sqrt{15}-8-2\sqrt{15}}{2}}$

$\leadsto \mathrm{x-y=\dfrac{-4\sqrt{15}}{2}}$

$\leadsto \mathrm{x-y=-2\sqrt{15}}$

_____________

$\leadsto \mathrm{xy=\dfrac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}\times\dfrac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}}$

$\leadsto \mathrm{xy=1}$

Now,

$\leadsto \sf{\dfrac{x^{2}+xy+y^{2}}{x^{2}-xy+y^{2}}}$

$\leadsto \sf{\dfrac{x^{2}+xy+y^{2}+xy-xy}{x^{2}-xy+y^{2}+xy-xy}}$

$\leadsto \sf{\dfrac{x^{2}+2xy+y^{2}-xy}{x^{2}-2xy+y^{2}+xy}}$

$\leadsto \sf{\dfrac{(x+y)^{2}-xy}{(x-y)^{2}+xy}}$

Putting the values :-

$\leadsto \sf{\dfrac{(8)^{2}-1}{(-2\sqrt{15})^{2}+1}}$

$\leadsto \sf{\dfrac{64-1}{60+1}}$

$\leadsto \sf{\dfrac{63}{61}}$

Hence, the required value is  $\tt{\dfrac{63}{61}}$.

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$\large\qquad\star{\underline{\underline{\frak{\red{Need\;to\;Know :}}}}}$

•$\tt{(a+b)^{2}=a^{2}+2ab+b^{2}}$

•$\tt{(a-b)^{2}=a^{2}-2ab+b^{2}}$

•$\tt{(a+b)(a-b)=a^{2}-b^{2}}$

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