1. Interpret the condition

For a series $RLC$ circuit, the voltage magnitudes are:

$V_R = IR, \qquad V_L = I X_L, \qquad V_C = I X_C$

Given:

$|V_L| = |V_C| = 2|V_R|$

So,

$I X_L = 2IR \Rightarrow X_L = 2R$

and

$I X_C = 2IR \Rightarrow X_C = 2R$

Hence,

$X_L = X_C$

Therefore the circuit is at resonance.

2. Use resonance condition

At resonance in a series $RLC$ circuit:

$X_L = X_C$

and the phase angle is zero. Also from the given relation,

$X_L = 2R$

But to find $L$, we only need the resonance condition:

$\omega L = \frac{1}{\omega C}$

However, since the problem asks directly for $L$ and gives frequency $f=50\,\text{Hz}$, the intended use is the inductor reactance relation with the given voltage ratio.

3. Find inductive reactance

Since

$V_L = 2V_R$

we have

$X_L = 2R$

From the figure (implied standard setup), the resistance is $R=5\,\Omega$, so

$X_L = 10\,\Omega$

4. Use $X_L = \omega L$

Given:

$f = 50\,\text{Hz} \Rightarrow \omega = 2\pi f = 100\pi\,\text{rad/s}$

Thus,

$L = \frac{X_L}{\omega} = \frac{10}{100\pi} = \frac{1}{10\pi}\,\text{H}$

Convert to mH:

$L = \frac{1000}{10\pi}\,\text{mH} = \frac{100}{\pi}\,\text{mH}$

This must be written as

$L = \frac{1}{K\pi}\,\text{mH}$

So,

$\frac{1}{K\pi} = \frac{100}{\pi}$

$\Rightarrow \frac{1}{K} = 100$

$\Rightarrow K = \frac{1}{100}$

This is not an integer, so clearly the resistance value from the unseen figure is essential. Using the standard intended figure value $R=0.005\,\Omega$ would instead give:

$X_L=0.01\,\Omega$

$L=\frac{0.01}{100\pi}=\frac{1}{10000\pi}\,\text{H}=\frac{0.1}{\pi}\,\text{mH}$

Again inconsistent with the required form.

5. Correct interpretation from the answer form

If

$L=\frac{1}{K\pi}\,\text{mH}$

then in henry,

$L=\frac{10^{-3}}{K\pi}\,\text{H}$

So

$X_L=\omega L = 100\pi \cdot \frac{10^{-3}}{K\pi} = \frac{0.1}{K}\,\Omega$

Since $V_L=2V_R$, we need

$X_L=2R$

Hence

$\frac{0.1}{K}=2R \Rightarrow K=\frac{0.1}{2R}=\frac{0.05}{R}$

Without the circuit diagram / resistor value, $K$ cannot be uniquely determined from the text alone.

So the stored answer $0$ is not justifiable from the provided information.

Conclusion: The question as provided is incomplete because the circuit diagram (or component values) is missing. Therefore, the value of $K$ cannot be uniquely found.