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x-1-x-2-y-1-y-2-1-What-is-the-value-of-x-y-2-

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Question Number 191484 by MATHEMATICSAM last updated on 24/Apr/23
(x + (√(1 + x^2 )))(y + (√(1 + y^2 ))) = 1 What is the value of (x + y)^2 ?
(x+1+x2)(y+1+y2)=1Whatisthevalueof(x+y)2?
Answered by witcher3 last updated on 24/Apr/23
ln(x+(√(1+x^2 ))) +ln(y+(√(1+y^2 )))=0 ⇔y+(√(y^2 +1))=(1/(x+(√(1+x^2 ))))=(√(1+x^2 ))−x ⇒let f(t)=y+(√(1+y^2 )) f′(t)=1+(y/( (√(1+y^2 ))))>0....E we have f(y)=f(−x)⇒ withe E y=−x
ln(x+1+x2)+ln(y+1+y2)=0⇔y+y2+1=1x+1+x2=1+x2−x⇒letf(t)=y+1+y2f′(t)=1+y1+y2>0….Ewehavef(y)=f(−x)⇒witheEy=−x
Answered by mr W last updated on 24/Apr/23
x+(√(1+x^2 ))=(√(1+y^2 ))−y x+(√(1+x^2 ))=−y+(√(1+(−y)^2 )) ⇒x=−y ⇒x+y=0 ⇒(x+y)^2 =0
x+1+x2=1+y2−yx+1+x2=−y+1+(−y)2⇒x=−y⇒x+y=0⇒(x+y)2=0

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