طول رأس سهمی $ = 2 = \frac{{ - a}}{{2b}} \Rightarrow 4b + a = 0$

رأس سهمی $ = (2,1) \Rightarrow 1 = 4b + 2a - 1 \Rightarrow 4b + 2a = 2$

$\left\{ \begin{gathered}
  4b + a = 0 \hfill \\
  4b + 2a = 2 \hfill \\ 
\end{gathered}  \right. \Rightarrow a = 2\,,\,b = \frac{{ - 1}}{2}$

$f(x) \Rightarrow y = \frac{{ - 1}}{2}{x^2} + 2x - 1 \Rightarrow \frac{{ - 1}}{2}{x^2} + 2x - 1 = 0 \to \left\{ \begin{gathered}
  \alpha  = 2 + \sqrt 2  \hfill \\
  \alpha  = 2 - \sqrt 2  \hfill \\ 
\end{gathered}  \right.$

$ \Rightarrow g(x) = {x^2} + (2 + \sqrt 2 )x + k \Rightarrow \Delta  \lt 0 \to {(2 + \sqrt 2 )^2} - 4(1)(k) \lt 0$

$6 + 4\sqrt 2  - 4k \lt 0 \Rightarrow 6 + 4\sqrt 2  \lt 4k \Rightarrow \frac{{6 + 4\sqrt 2 }}{4} \lt k$

$2/9 \lt k \Rightarrow k \in Y$