Question

please help me in this question ​

Answer 1

$\blue{\bold{\underline{\underline{Answer:}}}}$

$\green{\tt{\therefore{x^{2}+y^{2}=98}}}$

$\orange{\bold{\underline{\underline{Step-by-step\:explanation:}}}}$

$\green{\underline \bold{Given : }} \\ \tt: \implies x = \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} } \\ \\ \tt: \implies y = \frac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} + \sqrt{2} } \\ \\ \red{\underline \bold{to \: find : }} \\ \tt: \implies {x}^{2} + {y}^{2} = ?$

• According to given question :

$\bold{Rationalising : } \\ \tt: \implies x = \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} } \times \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} + \sqrt{2} } \\ \\ \tt: \implies x = \frac{ (\sqrt{3} + \sqrt{2} )^{2} }{ {( \sqrt{3}})^{2} - { (\sqrt{2} })^{2} } \\ \\ \tt: \implies x = \frac{ {( \sqrt{3} )}^{2} + {( \sqrt{2} )}^{2} + 2 \sqrt{6} }{3 - 2} \\ \\ \tt: \implies x =3 + 2 + 2 \sqrt{6} \\ \\ \tt: \implies x =5 + 2 \sqrt{6} \\ \\ \bold{Similarly : } \\ \tt: \implies y = \frac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} + \sqrt{2} } \times \frac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} - \sqrt{2} }$

$\tt: \implies y = \frac{ (\sqrt{3} - \sqrt{2})^{2} }{ {( \sqrt{3} )}^{2} - { (\sqrt{2} })^{2} } \\ \\ \tt: \implies y= \frac{ { (\sqrt{3} })^{2} + {( \sqrt{2} )}^{2} - 2 \sqrt{6} }{3 - 2 } \\ \\ \tt: \implies y=3 + 2 - 2 \sqrt{6} \\ \\ \tt: \implies y =5 - 2 \sqrt{6} \\ \\ \bold{For \: finding \: value : } \\ \tt: \implies {x}^{2} + {y}^{2} =(x + y)^{2} - 2xy \\ \\ \tt: \implies {x}^{2} + {y}^{2} =(5 + 2 \sqrt{6} + 5 - 2 \sqrt{6} )^{2} - 2(5 + 2\sqrt{6} )(5 - 2 \sqrt{6} ) \\ \\ \tt: \implies {x}^{2} + {y}^{2} = {10}^{2} - 2( {5}^{2} - {(2 \sqrt{6}) }^{2} \\ \\ \tt: \implies {x}^{2} + {y}^{2} =100 - 2(25 - 24) \\ \\ \tt: \implies {x}^{2} + {y}^{2} =100 - 2 \\ \\ \green{\tt: \implies {x}^{2} + {y}^{2} =98}$

Answer 2

⠀⠀⠀⠀⠀$\huge{ \color{purple}{ \bold{ \bf{AnSweR }}}}$

x² + y ² = 98

⠀⠀⠀$\huge{ \underline{ \purple{ \bold{ \underline{ \mathrm{ExPlanA{\green{TiOn }}}}}}}}$

$\large\underline{ \underline{ \color{green}{ \bold {Given}}}}$

➥ $\bf\:x= \frac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} + \sqrt{2} }$

➥ $\bf\:x= \frac{ \sqrt{ 3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} }$

$\large\underline{ \underline{ \color{green}{ \bold {To \:Find}}}}$

we need to find the Value of x²+ y²

⠀⠀⠀⠀⠀$\huge\underline{ \underline{ \color{red}{ \bold{sOluTiOn}}}}$

⠀⠀⠀⠀⠀

Rationalising the denominators of x and y values.

$</strong>\implies<strong> \bf\:x= \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} - \sqrt{2} } \times \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3 } + \sqrt{2} } \\ \\ </strong><strong>\implies</strong><strong> \bf \: x = \frac{ ({ \sqrt{3}) }^{2} +( { \sqrt{2} }^{2}) + 2 \times \sqrt{3} \times \sqrt{2} }{( \sqrt{3}) {}^{2} - (\sqrt{2} ) {}^{2} } \\ \\ </strong><strong>\implies</strong><strong> \bf \: x = \frac{3 + 2 + 2 \times \sqrt{6} }{3 - 2} \\ \\</strong>\implies<strong> \bf \: x = \frac{5 + 2 \sqrt{6} }{1}$

$\bf\:y = \frac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} + \sqrt{2} } \times \frac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} - \sqrt{2} } \\ \\ \bf \: y = \frac{ (\sqrt{3}) {}^{2} + { (\sqrt{2} )}^{2} - 2 \times \sqrt{3 } \times \sqrt{2} }{{ (\sqrt{3} ) {}^{2} - ( \sqrt{2} ){}^{2} } } \\ \\ \bf \: y = \frac{3 + 2 - 2 \times \sqrt{6} }{3 - 2} \\ \\ \bf \: y = \frac{5- 2 \sqrt{6} }{1}$

Now,

x² + y² = $\bf\:(5 + 2 \sqrt{6} ) {}^{2} + (5 - 2 \sqrt{6}){}^{2}$

x² + y² = $</strong><strong>\</strong><strong>s</strong><strong>m</strong><strong>a</strong><strong>l</strong><strong>l</strong><strong> \bf \: 25 + 24 + 20 \sqrt{6} + 25 + 24 - 20 \sqrt{6}$

x² + y² =$\bf \: 49 + 49$

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀= 98

⠀

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