Answer
See the detailed answer below.
Work Step by Step
$$\color{blue}{\bf [a]}$$
If you know that the electric field at the center of a charged disk is $\eta/2\epsilon_0$ and that the electric field at some point that is at a distance of $z$ on the disk's axis is just half this value. Find $z$ in terms of the radius of the disk $R$.
$$\color{blue}{\bf [b]}$$
Solving the given formula for $z$, we can see that $\eta/2\epsilon_0$ cancels from both sides,
$$1-\dfrac{z}{\sqrt{z^2+R^2}}=\dfrac{1}{2}$$
Hence,
$$2z= \sqrt{z^2+R^2} $$
Squaring both sides,
$$4z^2= z^2+R^2 $$
$$3z^2= R^2$$
$$z=\sqrt{\dfrac{ R^2}{3}}$$
$$\boxed{z=\dfrac{ R}{\sqrt3}}$$
