1. Given data

• Supply voltage: $V_s = 50\,\text{V}$
• Zener voltage: $V_Z = 5\,\text{V}$
• Maximum zener current: $I_{Z,\max} = 90\,\text{mA} = 0.09\,\text{A}$

2. Voltage across the series resistor

Since the zener diode maintains a voltage of $5\,\text{V}$ across itself, the remaining voltage appears across the resistor:

$V_R = V_s - V_Z = 50 - 5 = 45\,\text{V}$

3. Condition for minimum resistance

To ensure the zener current does not exceed its maximum value, the resistor current should be at most $0.09\,\text{A}$.

Using Ohm's law:

$R_{\min} = \frac{V_R}{I_{Z,\max}} = \frac{45}{0.09}$

4. Calculation

$R_{\min} = 500\,\Omega$

5. Final answer

The minimum required resistance is:

$\boxed{500\,\Omega}$