A potential divider circuit is connected with a dc source of 20 V, a light emitting diode of glow in voltage 1.8 V and a zener diode of breakdown voltage of 3.2 V. The length (PR) of the resistive wire is 20 cm. The minimum length of PQ to just glow the LED is _____ cm

This is a potentiometer (potential divider) setup.

Total length PR = 20 cm
Total voltage = 20 V

So potential gradient:

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

step 1: condition for LED to just glow

LED forward voltage = 1.8 V
Zener breakdown = 3.2 V

Both are in series (in conduction path), so total required:

$$V=1.8+3.2=5\text{ V}$$

step 2: required length

$$l=\frac{V}{k}=\frac{5}{1}=5\text{ cm}$$