Given-x-x-2-1-y-y-2-4-9-find-the-value-of-x-y-2-4-y-x-2-1-

Question Number 138150 by liberty last updated on 10/Apr/21

G i v e n (x + \sqrt{x^{2} + 1}) (y + \sqrt{y^{2} + 4}) = 9

f i n d t h e v a l u e o f

x \sqrt{y^{2} + 4} + y \sqrt{x^{2} + 1} .

Answered by EDWIN88 last updated on 10/Apr/21

s e t {\begin{cases} x = \cot 2 α \\ y = 2 \cot 2 β \end{cases} w i t h 0 < α, β < π / 2

w e g e t \begin{array}{cc} x + \sqrt{x^{2} + 1} = \frac{\cos 2 α}{\sin 2 α} + \frac{1}{\sin 2 α} = \cot α \\ y + \sqrt{y^{2} + 4} = \frac{2 \cos 2 β}{\sin 2 β} + \frac{2}{\sin 2 β} = 2 \cot β \end{array}

t h e n f r o m t h e g i v e n e q u a t i o n

2 \cot α . \cot β = a \frac{\cos α \cos β}{\sin α \sin β} = \frac{a}{2}

w e w a n t t o c o m p u t e x \sqrt{y^{2} + 4} + y \sqrt{x^{2} + 1} =

\cot 2 α (\frac{2}{\sin 2 β}) + 2 \cot 2 β (\frac{1}{\sin 2 α}) =

\frac{2 (\cos 2 α + \cos 2 β)}{\sin 2 α \sin 2 β} = \frac{\cos 2 α + \cos 2 β}{2 (\cos α \cos β) (\sin α \sin β)}

= \frac{2 - {2 \sin}^{2} α - 2 \sin^{2} β}{2 (\frac{a}{2}) \sin^{2} α \sin^{2} β}

= \frac{2}{a} (\frac{1 - \sin^{2} α - \sin^{2} β}{\sin^{2} α \sin^{2} β})

= \frac{2}{a} ((1 + \cot^{2} α) (1 + \cot^{2} β) - 1 - \cot^{2} α - 1 - \cot^{2} β)

= \frac{2}{a} (\cot^{2} α \cot^{2} β - 1)

= \frac{2}{a} (\frac{a^{2}}{4} - 1) = \frac{2}{a} (\frac{a^{2} - 4}{4}) = \frac{a^{2} - 4}{2 a}

i n t h i s c a s e a = 9; w e g e t \frac{81 - 4}{18} = \frac{77}{18}

Commented by liberty last updated on 11/Apr/21

n i c e

Answered by MJS_new last updated on 10/Apr/21

u = x + \sqrt{x^{2} + 1} \Leftrightarrow x = \frac{u^{2} - 1}{2 u}

v = y + \sqrt{y^{2} + 4} \Leftrightarrow y = \frac{v^{2} - 4}{2 v}

v = \frac{9}{u} \Rightarrow y = \frac{81 - 4 u^{2}}{18 u}

x \sqrt{y^{2} + 4} + y \sqrt{x^{2} + 1} = \dots = \frac{77}{18}

Commented by liberty last updated on 11/Apr/21

s h o r t c u t ?

Commented by MJS_new last updated on 11/Apr/21

well I guess {it}^{'} s not necessary to type all the

algebraic steps \dots

btw . we can also solve the given equation

for y :

X (y + \sqrt{y^{2} + 4}) = 9 \Rightarrow y = \frac{81 - 4 X^{2}}{18 X} = \frac{- 85 x + 77 \sqrt{x^{2} + 1}}{18}

\Rightarrow x \sqrt{y^{2} + 4} + y \sqrt{x^{2} + 1} = \dots = \frac{77}{18}

Commented by SLVR last updated on 11/Apr/21

t h i s p a r t o f s o l u t i o n i s m a r v o l e s

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