Question

Find the equivalent resistance of the infinite ladder circuit shown in figure across terminal <!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> $\text{A}$ and $\text{B}$.

• A. $\frac{1+\sqrt{11}}{2}\Omega$
• B. $\frac{2+\sqrt{22}}{2}\Omega$
• C. $\frac{3+\sqrt{33}}{2}\Omega$
• D. $\frac{8+\sqrt{39}}{5}\Omega$

Answer 1

The correct option is C $\frac{3+\sqrt{33}}{2}\Omega$

If equivalent resistance across terminals <!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> $\text{A}$ and $\text{B}$ is considered as $R_0$ then across points <!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> $\text{C}$ and $\text{D}$ toward right of it, the network resistance will also be the same.

Thus, we can reduce the given circuit as shown in figure.


Using series and parallel combination, the equivalent resistance of this circuit is given as,

$R_0=3+\frac{2R_0}{2+R_0}$

$\Rightarrow 2R_0+R_0^2=6+5R_0$

$\Rightarrow R_0^2-3R_0-6=0$

$\Rightarrow R_0=\frac{3\pm \sqrt{9+24}}{2}$

As, magnitude of resistance cannot be negative.

$\therefore R_0=\frac{3+\sqrt{33}}{2}\Omega$

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> Hence, $\left(c\right)$ is the correct option.