Which of the following represents a point in an Argand plane, equidistant from the roots of the equation $(\mathrm{z}+1{)}^4=16{\mathrm{z}}^4$?

$\begin{array}{c}{\left(\frac{\mathrm{z}+1}{\mathrm{z}}\right)}^4=16\\ \Rightarrow \frac{\mathrm{z}+1}{\mathrm{z}}=\pm 2,\pm 2\mathrm{i}\end{array}$

The roots are 1, -1/3, (-1/5 - (2/5)i), and (-1/5 + (2/5)i).

Note that (-1/3, 0) and (1, 0) are equidistant from (1/3, 0) and since it lies on the perpendicular bisector of AB, it will be equidistant from A and B also.