If `x^2+y^2-2x+2a y+a+3=0` represents the real circle with nonzero radius, then find the values of `a`.

Explanation

To determine the values of \( a \) such that the equation \( x^2 + y^2 - 2x + 2ay + a + 3 = 0 \) represents a real circle with a nonzero radius, we will follow these steps:

### Step 1: Identify the general form of the circle
The general equation of a circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
where \( g \), \( f \), and \( c \) are constants.
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