The combined resistance R of two resistors $R_1$ and $R_2(R_1, R_2 > 0)$ is given by $\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}$ if $R_1+R_2=C$ (a constant). The maximum resistance R is obtained when :
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → $R_1=R_2$
$\frac{1}{R}=\frac{1}{R_1}+\frac{1}{R_2}⇒R=\frac{R_1R_2}{R_1+R_2}=\frac{R_1(C-R_1)}{C}$
$\frac{dR}{dR_1}=\frac{C-R_1}{C}-\frac{R_1}{C}=0$
$⇒R_1=\frac{C}{2}$ so $R_2=\frac{C}{2}$
$\frac{d^2R}{d{R_1}^2}=-\frac{2}{C}<0$ (at $R_1=\frac{C}{2}$ maxima occurs)
so $(R_1=R_2)$ ⇒ maximum resistance