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The solution to the system of equations gives:

- \(k = -\frac{1}{2}\)
- \(r^2 = \frac{25}{4}\)
- \(h = -1\) (since \(h = 2k\))

Thus, the center of the circle is \((-1, -\frac{1}{2})\) and the radius squared is \(\frac{25}{4}\).

The equation of the circle is:

\[
(x + 1)^2 + \left(y + \frac{1}{2}\right)^2 = \frac{25}{4}
\]