if-1-x-2-x-1-y-2-y-1-with-x-y-R-find-x-y-2-

Question Number 189022 by mr W last updated on 10/Mar/23

i f (\sqrt{1 + x^{2}} + x) (\sqrt{1 + y^{2}} + y) = 1

w i t h x, y \in R, f i n d {(x + y)}^{2} = ?

Answered by Frix last updated on 11/Mar/23

Obviously y = - x \Rightarrow answer is 0

[\sqrt{x^{2} + 1} + x = \frac{1}{\sqrt{x^{2} + 1} - x}]

Commented by mr W last updated on 11/Mar/23

c a n w e b e s u r e t h a t n o o t h e r s o l u t i o n s

e x i s t ?

Answered by Frix last updated on 11/Mar/23

\sqrt{x^{2} + 1} + x = \frac{1}{\sqrt{y^{2} + 1} + y}

\sqrt{x^{2} + 1} + x = \sqrt{y^{2} + 1} - y

x + y = \sqrt{y^{2} + 1} - \sqrt{x^{2} + 1}

x^{2} + 2 x y + y^{2} = y^{2} + 1 + x^{2} + 1 - 2 \sqrt{x^{2} + 1} \sqrt{y^{2} + 1}

x y = 1 - \sqrt{x^{2} + 1} \sqrt{y^{2} + 1}

1 - x y = \sqrt{x^{2} + 1} \sqrt{y^{2} + 1}

1 - 2 x y + x^{2} y^{2} = x^{2} y^{2} + x^{2} + y^{2} + 1

- 2 x y = x^{2} + y^{2}

{(x + y)}^{2} = 0

Commented by mr W last updated on 11/Mar/23

n i c e s o l u t i o n, t h a n k s!

Answered by MJS_new last updated on 11/Mar/23

z = \sinh u \land y = \sinh v

e^{u} e^{v} = 1

e^{u + v} = 1

u + v = 0

v = - u

\sinh (- u) = - \sinh u

y = - x

Commented by mr W last updated on 11/Mar/23

n i c e s o l u t i o n, t h a n k s!

Facebook Tweet Pin