121 Let $p \in \mathbf{R}$, in the complex plane, denote the two complex roots of the equation
$$
x^{2}-2 x+2=0
$$

by the complex number points $A, B$; denote the two complex roots of the equation
$$
x^{2}+2 p x-1=0
$$

by the complex number points $C, D$. If $A, B, C, D$ are concyclic, then $p=$
$\qquad$ .
$121-\frac{1}{2}$. Since $A, B$ are $1+i$ and $1-i$, and $C, D$ are $-p+\sqrt{p^{2}+1}$ and $-p-\sqrt{p^{2}+1}$.
$CD$ is the perpendicular bisector of $AB$, and the four points are concyclic, with the center obviously being the midpoint of $CD$ $(-p, 0)$. The distance from this point to $1+i$ and $-p+\sqrt{p^{2}+1}$ is equal, i.e.,
$$
p^{2}+1=(1+p)^{2}+1^{2}
$$

Thus, $p=-\frac{1}{2}$.