Q:

The maximum value of r?

A point  $A = (a,b)$ is defined such that it lies on the graph $y = x^2 +1$
A point $B = (c,d)$ is defined such that it lies WITHIN the area of $ (x+2)^2 + (y+2)^2 = r^2$
Let's define a matrix $M = \begin{pmatrix}
a &b \\ 
 c& d
\end{pmatrix}$ so that M always has an inverse. (i.e. determinant does not equal 0)
What is the maximum possible $50r^2$? 

Apparently this is a highschool math problem, but I just can't solve it. I feel ashamed now. 

A:

I've enquired in the comment and you seem to think this is the correct question, so let's analyze it.
You have $a=b^2+1$, $(c+2)^2+(d+2)^2\leq r^2$, and $ad-bc\neq0$. Let $d=0$, $b=1$, $a=2$, then $c$ may take any value except zero, thus this imposes no restriction on $r^2$.

