Explain why $\sqrt{x^2-y^2}+\arcsin(x/y)=0$ does not define $y$ as an implicit function of $x$.
Quite confused by this, mainly because I do not fully understand really what it means for an equation to define $y$ as an implicit function of $x$ even though I kind of get the idea.
I think the reason is something to do with the fact that the first term requires $|x|\ge|y|$ and the second term the opposite but I would appreciate if someone would explain this to me a bit more clearly, thanks.
Edit: Sorry I messed up the question it was meant to be a minus in the root.
The question sounds a little strange to me… But I agree with what you said in a comment above — essentially, the reason is that this equation has no solutions at all, so it doesn't define anything. Geometrically, its graph is the empty set in the plane.
And yes, just as you said, it has a lot to do with the fact that the first term requires $|x|\ge|y|$ and the second term requires $|x|\le|y|$, along with $y\ne0$. The two inequalities together imply that $|x|=|y|$, so $y=\pm x$. In either case, $\sqrt{x^2-y^2}=0$, so we're left with $\arcsin(x/y)=\arcsin(\pm1)=\pm\pi/2\ne0$.