The vector $\overrightarrow a=\alpha\hat i+2\hat j+\beta\hat k$ lies in the plane of the vectors $\overrightarrow b=\hat i+\hat j$ and $\overrightarrow c=\hat j+\hat k$ and bisects the angle between $\overrightarrow b\:\:and\:\:\overrightarrow c$, then $(\alpha,\beta)=?$

Question

$(a)\:\:(1,1)\:\:\:\:\qquad\:\:(b)\:\:(2,2)\:\:\:\:\qquad\:\:(c)\:\:(1,2)\:\:\:\:\qquad\:\:(d)\:\:(2,1).$