1. Potential divider along the wire

A uniform resistive wire $PR$ of length $20\,\text{cm}$ is connected across a dc source of $20\,\text{V}$.

Hence, the potential drop per unit length is

$\frac{20\,\text{V}}{20\,\text{cm}}=1\,\text{V/cm}.$

So, if the jockey/contact is at point $Q$ such that $PQ=x$, then the potential difference across $PQ$ is

$V_{PQ}=x\,\text{V}.$

2. Condition for LED to just glow

The LED has glow voltage $1.8\,\text{V}$. The zener diode has breakdown voltage $3.2\,\text{V}$.

For the LED to just glow in this arrangement, the required minimum potential difference across the selected portion must be enough to account for both thresholds:

$V_{PQ}=V_{\text{LED}}+V_Z=1.8+3.2=5.0\,\text{V}.$

3. Convert required voltage into length

Since the wire provides $1\,\text{V}$ per cm,

$PQ=5.0\,\text{cm}.$

4. Final answer

Therefore, the minimum length of $PQ$ required to just glow the LED is

$\boxed{5\,\text{cm}}.$