Question

If $x,y,z$ satisfies the equations
$x+\log (x+\sqrt{x^2+1})=y$
$y+\log (y+\sqrt{y^2+1})=z$
$z+\log (z+\sqrt{z^2+1})=x$, then find the value of ${(1-x)}^2+{(1-y)}^2+{(1-z)}^2$.

Answer 1

If x , y , z satisfies the equations:

$x+\log (x+\sqrt{x^2+1})=yy+\log (y+\sqrt{y^2+1})=zz+\log (z+\sqrt{z^2+1})=x$

The equations are in cyclic form hence x = 0 , y = 0 , z = 0 ratifies equation and x = y = z satisfies these equations.

putting values of x , y , z in equation given:

${(1-x)}^2+{(1-y)}^2+{(1-z)}^2=(1-0{)}^2+(1-0{)}^2+(1-0{)}^2=1+1+1=3$

Ans = 3