An AC circuit has an inductor and a resistor of resistance $$R$$ in series, such that $$X_L = 3R$$. Now, a capacitor is added in series such that $$X_C = 2R$$. The ratio of the new power factor with the old power factor of the circuit is $$\sqrt{5} : x$$. The value of $$x$$ is _________.

We begin with the original series circuit that contains only a resistor of resistance $$R$$ and an inductor whose inductive reactance is $$X_L = 3R$$. In any series $$R$$-$$L$$ circuit, the impedance is given by the formula

$$Z = \sqrt{R^{2} + X_L^{2}}.$$

Substituting $$X_L = 3R$$, we obtain

$$Z_{\text{old}} = \sqrt{R^{2} + (3R)^{2}} = \sqrt{R^{2} + 9R^{2}} = \sqrt{10R^{2}} = R\sqrt{10}.$$

The power factor of an AC circuit is the cosine of the phase angle between the voltage and the current, and in a series circuit it is given by

$$\cos\phi = \frac{R}{Z}.$$

So, for the original circuit,

$$\cos\phi_{\text{old}} = \frac{R}{R\sqrt{10}} = \frac{1}{\sqrt{10}}.$$

Now a capacitor is connected in series, having a capacitive reactance $$X_C = 2R$$. In a series $$R$$-$$L$$-$$C$$ circuit, the net reactance is the difference between the inductive and capacitive reactances:

$$X = X_L - X_C.$$

Substituting the given values, we get

$$X = 3R - 2R = R.$$

Using the impedance formula again, this time with the new net reactance $$X = R$$, we have

$$Z_{\text{new}} = \sqrt{R^{2} + X^{2}} = \sqrt{R^{2} + R^{2}} = \sqrt{2R^{2}} = R\sqrt{2}.$$

The new power factor is therefore

$$\cos\phi_{\text{new}} = \frac{R}{R\sqrt{2}} = \frac{1}{\sqrt{2}}.$$

We are asked for the ratio of the new power factor to the old power factor. Writing this ratio explicitly, we have

$$\frac{\cos\phi_{\text{new}}}{\cos\phi_{\text{old}}} = \frac{\dfrac{1}{\sqrt{2}}}{\dfrac{1}{\sqrt{10}}} = \frac{1}{\sqrt{2}}\times\frac{\sqrt{10}}{1} = \sqrt{\frac{10}{2}} = \sqrt{5}.$$

Hence,

$$\cos\phi_{\text{new}} : \cos\phi_{\text{old}} = \sqrt{5} : 1.$$

Comparing with the statement in the question, $$\sqrt{5} : x$$, we immediately identify $$x = 1.$$

Hence, the correct answer is Option 1.