Simply said, radius \(r\) can go to \(\infty\).

\((4,0)\) and \((-4,0)\) lie of the a circle, which means the circle is symmetric about y-axis

Let \((h,k)\) be the center of the circle.
Since the circle is symmetric about y-axis, it makes \(h = 0\)

Equation of this circle -> \((x-h)^2 + (y-k)^2 = r^2\)

Now \((4,0)\) and \((-4,0)\) lie of the a circle.
\((4-0)^2 + (0-k)^2 = r^2\) and \((-4-0)^2 + (0-k)^2 = r^2\)

\((4)^2 + (k)^2 = r^2\)

\(4^2 + k^2 = r^2\)

\(r\) can be interpreted as hypotenuse of a right-angles triangle, considering the Pythagorean Theorem into consideration.

But here is the catch, \(r\) is not restricted to be an integer and hence its value can literally go to infinity.

Hence, Answer is E