I have a query below, and want to understand the reason of this in detail.
Here is the graph of $x^2+(y-\sqrt{x^2})^2=1$ drawn by Wolfram Alpha:
x^2+(y-\sqrt{x^2})^2=1">
In this equation if you replace the term, $\sqrt{x^2}$ by $x$, which is a valid substitution, the graph seems to become an ellipse:
x^2+(y-x)^2=1">
Shouldn't the curve remain same? What is the reason for this? Am I seriously missing some fundamental point here?