Question

For what value of the resistor $X$ will the equivalent resistance of the two circuits shown be the same?

• A. $R$
• B. $6R$
• C. $R$
• D. $\frac{\sqrt{5-1}}{2}R$

Answer 1

Answer: (C)

Given circuits are Circuit $I$,

Circuit $II,$

Clearly, both circuits are different after sections $AB$ and $CD.$
So, resistances of two circuits are same, if
$R_{AB}=R_{CD}$
So, $R+X=R+\frac{6R\left(R+X\right)}{6R+R+X}$
$\Rightarrow X=\frac{6R\left(R+X\right)}{7R+X}$
$\Rightarrow 7RX+X^2=6X^2+6RX$
$\Rightarrow X^2+RX-6R^2=0$
From sridharacharya formula, we have
$\Rightarrow X=\frac{-R\pm \sqrt{R^2-4\left(1\right)\left(-6R^2\right)}}{2\left(1\right)}$
$=\frac{-R\pm \sqrt{25R^2}}{2}$
or $X=\frac{-R\pm 5R}{2}$
$\therefore X=2R$