Complex Number solutions for a Circle I have a circle of radius 5 with its center at the origin represented as $X^2+Y^2=25$. I get that it has a solution for all values ranging from $-5$ to $+5$.

My question is what does it mean when the equation returns a complex number? Example for $x=6$, I get $y = \pm i\sqrt{11}$. In doing this for all real number greater than $+5$ and less than $-5$, what is being returned/plotted and what plane is this plot on? Is this another circle on the imaginary plane, all values on the plane beyond a circle?
When you graph the solutions in "the plane", you are restricting yourself to look at solutions to the equation where both $x$ and $y$ are real.  You could, for example, restrict further to only allow $x$ and $y$ to be rational numbers, and think about how those points fit in with all the real solutions.  

To think geometrically about non-real complex solutions, you will need more (real) dimensions!  You could restrict yourself, as it sounds like you are doing in the question, to just solutions where $x$ is real and $y$ is allowed to be complex.  Then you will need another dimension/direction for the imaginary part of $y$.  You could graph this in a "$z$" direction, so that solutions where $x&gt;5$ not lie in the plane but above/below it.  You will find that for $|x|&gt;5$ the solutions will be points where $x^2 - z^2 = 25$ so that if you just look at the $(x,z)$ plane the solution set will look like a hyperbola.  

Probably the most interesting thing to look at is when you allow both $x$ and $y$ to be complex... but graphing this would require more dimensions.
All the points that lie inside or on the boundary of the circle 
$$ \left\{ \ (x, y) \in \mathbb{R}^2 \ \colon \ x^2+ y^2 = 5^2 \ \right\} $$
lie to the right of the line 
$$ \left\{ \ (x, y) \in \mathbb{R}^2 \ \colon \  x = -5  \ \right\} $$
and to the left of the line
$$ \left\{ \ (x, y) \in \mathbb{R}^2 \ \colon \  x = 5  \ \right\}. $$
Moreover, the circle intersects the former line at the point $(-5, 0)$ and the latter one at the point $(5, 0)$.