Find the complete set of values of $p$ for which the equation $$x ^ { 2 } - 2 p x + y ^ { 2 } - 6 y - p ^ { 2 } + 8 p + 9 = 0$$ describes a circle in the $x y$-plane.
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Find the complete set of values of $p$ for which the equation
$$x ^ { 2 } - 2 p x + y ^ { 2 } - 6 y - p ^ { 2 } + 8 p + 9 = 0$$
describes a circle in the $x y$-plane.