\[y = - 2x^{2} + bx + c;\ \ A(2;1).\]
\[1)\ x_{0} = - \frac{b}{2a}\]
\[2 = - \frac{b}{- 4}\]
\[2 = \frac{b}{4}\]
\[b = 8.\]
\[2) - 2 \cdot 2^{2} + 8 \cdot 2 + c = 1\]
\[- 8 + 16 + c = 1\]
\[c = 1 - 8\]
\[c = - 7.\]
\[Ответ:при\ b = 8;\ \ c = - 7.\]