I4.2 Let $x$ and $y$ be real numbers satisfying the equation $y^{2}+4 y+4+\sqrt{x+y+k}=0$. If $r=|x y|$, find the value of $r$.
Reference: 2005 FI4.1, 2009 FG1.4, 2011 FI4.3, 2013 FI1.4, 2015 HG4, 2015 FI1.1
The equation is equivalent to $(y+2)^{2}+\sqrt{x+y+k}=0$
which is a sum of two non-negative numbers.
$$
\begin{array}{l}
\Rightarrow y+2=0 \text { and } x+y+1=0 \\
y=-2 \text { and } x-2+1=0, x=1 \\
r=|-2 \times 1|=2
\end{array}
$$