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Question Number 46944 by Tawa1 last updated on 02/Nov/18

Commented by maxmathsup by imad last updated on 03/Nov/18

$w e h a v e \frac{x}{y} + \frac{y}{x} = \frac{x^{2} + y^{2}}{x y} a n d$ $\frac{x}{s i n θ} = \frac{y}{c o s θ} \Rightarrow \frac{x^{4}}{{s i n}^{4} θ} = \frac{y^{4}}{{c o s}^{4} θ} (\times \frac{1}{x^{4} y^{4}}) \Rightarrow \frac{1}{y^{4} {s i n}^{4} θ} = \frac{1}{x^{4} {c o s}^{4}} = \frac{2}{x^{4} {c o s}^{4} θ + y^{4} {s i n}^{4} θ}$ $s o i f w e t a k e t h e c o n d . \frac{{s i n}^{4} θ}{x^{4}} + \frac{{c o s}^{4} θ}{y^{4}} = 97 \Rightarrow \frac{y^{4} {s i n}^{4} θ + x^{4} {c o s}^{4} θ}{x^{4} y^{4}} = 97 \Rightarrow$ $\frac{1}{y^{4} {s i n}^{4} θ} = \frac{2}{97 x^{4} y^{4}} \Rightarrow \frac{1}{{s i n}^{4} θ} = \frac{2}{97 y^{4}} \Rightarrow 97 y^{4} = 2 {s i n}^{4} θ a l s o w e g e t 97 x^{4} = 2 {c o s}^{4} θ \Rightarrow$ $97 (x^{4} + y^{4}) = 2 ({c o s}^{4} θ + {s i n}^{4} θ) \Rightarrow \frac{x^{4} + y^{4}}{{c o s}^{4} θ + {s i n}^{4} θ} = \frac{2}{97} b u t$ $\frac{x}{s i n θ} = \frac{y}{c o s θ} \Rightarrow \frac{x^{4}}{{s i n}^{4} θ} = \frac{y^{4}}{{c o s}^{4} θ} = \frac{x^{4} + y^{4}}{{s i n}^{4} θ + {c o s}^{4} θ} = \frac{2}{97} \Rightarrow$ $x^{4} = \frac{2}{97} {s i n}^{4} θ a n d y^{4} = \frac{2}{97} {c o s}^{4} θ \Rightarrow x^{2} = \sqrt{\frac{2}{97}} {s i n}^{2} θ a n d y^{2} = \sqrt{\frac{2}{97}} {c o s}^{2} θ$ $\Rightarrow x^{2} + y^{2} = \sqrt{\frac{2}{97}} b u t x = k s i n θ a n d y = k c o s θ \Rightarrow k^{2} = \sqrt{\frac{2}{97}} \Rightarrow$ $k =^{4} \sqrt{\frac{2}{97}} \Rightarrow \frac{x^{2} + y^{2}}{x y} = \frac{\sqrt{\frac{2}{97}}}{(^{4} \sqrt{\frac{2}{97}}) (^{4} \sqrt{\frac{2}{97}}) s i n θ c o s θ} = \frac{1}{s i n θ c o s θ}$
Commented by Tawa1 last updated on 03/Nov/18

$God bless you sir$
Commented by maxmathsup by imad last updated on 03/Nov/18

$y o u a r e w e l c o m e s i r .$
	Answered by tanmay.chaudhury50@gmail.com last updated on 03/Nov/18
	$((sinθ)/x)=((cosθ)/y)=(1/k) x=ksinθ y=kcosθ x^2 +y^2 =k^2 ((cos^4 θ)/(k^4 sin^4 θ))+((sin^4 θ)/(k^4 cos^4 θ))=97 cot^4 θ+tan^4 θ=97k^4 (cot^2 θ+tan^2 θ)^2 −2=97k^4 {(cotθ+tanθ)^2 −2}^2 =97k^4 +2 (cotθ+tanθ)^2 =2+(√(97k^4 +2)) cotθ+tanθ=[2+(√(97k^4 +2)) ]^(1/2) (y/x)+(x/y)=[2+(√(97k^4 +2)) ]^(1/2) (y/x)+(x/y)=[2+(√(97(x^2 +y^2 )^2 +2)) ]^(1/2)$
	$\frac{s i n θ}{x} = \frac{c o s θ}{y} = \frac{1}{k}$ $x = k s i n θ y = k c o s θ$ $x^{2} + y^{2} = k^{2}$ $\frac{{c o s}^{4} θ}{k^{4} {s i n}^{4} θ} + \frac{{s i n}^{4} θ}{k^{4} {c o s}^{4} θ} = 97$ ${c o t}^{4} θ + {t a n}^{4} θ = 97 k^{4}$ ${({c o t}^{2} θ + {t a n}^{2} θ)}^{2} - 2 = 97 k^{4}$ ${{(c o t θ + t a n θ)}^{2} - 2}^{2} = 97 k^{4} + 2$ ${(c o t θ + t a n θ)}^{2} = 2 + \sqrt{97 k^{4} + 2}$ $c o t θ + t a n θ = {[2 + \sqrt{97 k^{4} + 2}]}^{\frac{1}{2}}$ $\frac{y}{x} + \frac{x}{y} = {[2 + \sqrt{97 k^{4} + 2}]}^{\frac{1}{2}}$ $\frac{y}{x} + \frac{x}{y} = {[2 + \sqrt{97 {(x^{2} + y^{2})}^{2} + 2}]}^{\frac{1}{2}}$
	Commented by Tawa1 last updated on 03/Nov/18

	$God bless you sir$
	Commented by tanmay.chaudhury50@gmail.com last updated on 03/Nov/18

	$i s i t t h e a n s w e r . . .$
	Commented by Tawa1 last updated on 03/Nov/18

	$I {don}^{'} t know the answer too sir$
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