Confusing Bonus Question

Yes, Quadrant I provides the entire solution set for this question. Indeed for any given $c$, if we let $x=a$ and $y=b$, then $a+b=c^2$, translates to 
\[x+y=c^2  \Rightarrow  y=-x+c^2\]
which is a line with slope $m=-1$, and $x$-intercept and $y$-intercept are both $c^2$. 

For example for $c=5$ you get 
\[y=-x+25\]
which is the question of the red line in the image.