Let $f(x,y)=ln(\frac{x^2}{x^2+y^2}).$ Answer each of the following:

$($i$)$ Sketch the set of points in the domain;

$($ii$)$ Identify the range of $f(x,y)$;

$($iii$)$ Sketch the level curves for the levels $c=c=0,-1,-2,-\frac{1}{2}$;

$($iv$)$ Find an equation of the level curve passing through the point $(1,1).$

$($v$)$ Find an equation of the tangent line and an equation of the normal line to the level curve passing at $(1,1).$

$($vi$)$ Find an equation of the tangent plane and a parametric equation of the normal line to the surface $z=f(x,y)$ at $(1,1,-ln2).$

$($vii$)$ Determine if there is a point on the surface where the tangent plane is horizontal. If yes, find the point.