Let a = i +5 j +α k, b = i +3 j +β k and c =- i +2 j -3 k be three vectors such that, | b × c |=5 √ 3 and a is perpendicular to b. Then the greatest…

Let $\displaystyle \overrightarrow{\mathrm{a}}=\hat{i}+5 \hat{j}+\alpha \hat{\mathrm{k}}, \overrightarrow{\mathrm{b}}=\hat{i}+3 \hat{j}+\beta \hat{\mathrm{k}}$ and $\displaystyle \overrightarrow{\mathrm{c}}=-\hat{i}+2 \hat{j}-3 \hat{\mathrm{k}}$ be three vectors such that, $\displaystyle |\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}}|=5 \sqrt{3}$ and $\displaystyle \overrightarrow{\mathrm{a}}$ is perpendicular to $\displaystyle \overrightarrow{\mathrm{b}}$. Then the greatest amongst the values of $\displaystyle |\overrightarrow{\mathrm{a}}|^2$ is \_\_\_\_.