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:
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:
Shouldn't the curve remain same? What is the reason for this? Am I seriously missing some fundamental point here?
For each positive real number $y$, there are two numbers whose square is $y$, one positive and one negative. $\sqrt{y}$ is defined as the positive of those two numbers. The two numbers whose square is $x^2$ are $x$ and $−x$, so $\sqrt{x^2}$ is $x$ or $−x$, depending on which of those two is positive. If $x$ is negative, it is $−x$ that's positive, and thus $\sqrt{x^2}=−x$ for $x$ negative.