If $\mathrm{\alpha },\mathrm{\beta }$ are non-zero integers and $\mathrm{z}=(\mathrm{\alpha }+\mathrm{i\beta })(2+7\mathrm{i})$ is a purely imaginary number, then minimum value of $|\mathrm{z}{|}^2$ is

$\mathrm{z}=(\mathrm{\alpha }+\mathrm{i\beta })(2+7\mathrm{i})$

$$\begin{gathered} \mathrm{z}=2\alpha +2i\beta +7i\alpha -7\beta \\ z=\left(2\alpha -7\beta \right)+i\left(2\beta +7\alpha \right) \end{gathered}$$

purely imaginary then

$$\begin{gathered} \left(2\alpha -7\beta \right)=0 \\ 2\alpha =7\beta \\ \frac{\alpha }{\beta }=\frac{7}{2} \end{gathered}$$

$\alpha +i\beta =7+2i$

$$\begin{gathered} z=\left(7+2i\right)\left(2+7i\right) \\ z=53i \end{gathered}$$

$|\mathrm{z}{|}^2=2809$