The ratio of the equivalent resistance of the network (shown in figure) between the points $$a$$ and $$b$$ when switch is open and switch is closed is $$x : 8$$. The value of $$x$$ is _________.

We need to find the value of $$x$$ given that the ratio of the equivalent resistance of the network between points $$a$$ and $$b$$ when the switch is open to when the switch is closed is $$x : 8$$.

1. Case 1: When Switch $$S$$ is Open

When the switch $$S$$ is open, no current flows through the connecting branch. The circuit simplifies into two parallel branches:

• Top branch: Resistor $$R$$ and resistor $$2R$$ are connected in series.

  $$R_{\text{top}} = R + 2R = 3R$$

• Bottom branch: Resistor $$2R$$ and resistor $$R$$ are connected in series.

  $$R_{\text{bottom}} = 2R + R = 3R$$

Now, we find the equivalent resistance ($$R_{\text{open}}$$) of these two parallel branches ($$3R$$ and $$3R$$):

$$\frac{1}{R_{\text{open}}} = \frac{1}{3R} + \frac{1}{3R} = \frac{2}{3R} \implies R_{\text{open}} = \frac{3R}{2} = 1.5R$$

2. Case 2: When Switch $S$ is Closed

When the switch $$S$$ is closed, the central node connects the upper and lower paths together. The circuit transforms into a series combination of two parallel pairs:

• First parallel pair (left side): Resistors $$R$$ and $$2R$$ are in parallel.

  $$R_{\text{left}} = \frac{R \times 2R}{R + 2R} = \frac{2R^2}{3R} = \frac{2R}{3}$$

• Second parallel pair (right side): Resistors $$2R$$ and $$R$$ are in parallel.

  $$R_{\text{right}} = \frac{2R \times R}{2R + R} = \frac{2R^2}{3R} = \frac{2R}{3}$$

Now, we find the total equivalent resistance ($$R_{\text{closed}}$$) by adding the two series sections:

$$R_{\text{closed}} = R_{\text{left}} + R_{\text{right}} = \frac{2R}{3} + \frac{2R}{3} = \frac{4R}{3}$$

3. Find the Ratio and Solve for $x$

We are given that the ratio of the equivalent resistance is equal to $$x : 8$$:

$$\frac{R_{\text{open}}}{R_{\text{closed}}} = \frac{\frac{3R}{2}}{\frac{4R}{3}}$$

Simplify the fraction division:

$$\frac{R_{\text{open}}}{R_{\text{closed}}} = \frac{3}{2} \times \frac{3}{4} = \frac{9}{8}$$

Equating this to the given ratio:

$$\frac{x}{8} = \frac{9}{8} \implies x = 9$$