Even a small change in resistance $\backslash (\backslash$left$($R$\_\{$x$\}\backslash$right$)\backslash )$ causes the bridge to become unbalanced. These changes result in a non$-$zero voltage at $\backslash ($ V$\_\{\backslash$text $\{$out $\}\}\backslash )$ with the polarity indicating if $\backslash ($ R$\_\{$x$\}\backslash )$ is smaller or larger than the balanced value. Suppose Vin$=12$ V$,$ R$1=4$ Ohms, R$2=5$ Ohms and R$3$ is a variable resistor.

a$)$ Using Kirchoff's Voltage Law and Kirchoff's Current Law, demonstrate that R$1/$R$3=$ R$2/$Rx$.$

b$)$ Graph the power delivered to the Rx resistor versus the value of R $3$ resistor for Vout$=0$ volts. Find the value of R$3$ resistor which the power delivered to the Rx resistor is maximum.