Điện trở tương đương của mạch nối tiếp và song song là:
\(\left\{ \begin{align}
& {{R}_{nt}}={{R}_{1}}+{{R}_{2}} \\
& {{R}_{ss}}=\frac{{{R}_{1}}{{R}_{2}}}{{{R}_{1}}+{{R}_{2}}} \\
\end{align} \right.\)
Theo bài ra ta có:
\(\begin{align}
& {{R}_{nt}}=4,5{{R}_{ss}}\Rightarrow {{R}_{1}}+{{R}_{2}}=4,5\frac{{{R}_{1}}{{R}_{2}}}{{{R}_{1}}+{{R}_{2}}} \\
& \Leftrightarrow {{\left( {{R}_{1}}+{{R}_{2}} \right)}^{2}}=4,5{{R}_{1}}{{R}_{2}} \\
& \Leftrightarrow R_{1}^{2}+R_{2}^{2}=2,5{{R}_{1}}{{R}_{2}} \\
& \Leftrightarrow \frac{{{R}_{1}}}{{{R}_{2}}}+\frac{{{R}_{2}}}{{{R}_{1}}}=2,5\,\,\,\,\left( * \right) \\
\end{align}\)
Đặt :
\(\begin{array}{l}
\frac{{{R_1}}}{{{R_2}}} = x \Rightarrow \frac{{{R_2}}}{{{R_1}}} = \frac{1}{x}\\
\Rightarrow \left( * \right) \Leftrightarrow x + \frac{1}{x} = 2,5\\
\Rightarrow \left[ \begin{array}{l}
x = 0,5\\
x = 2
\end{array} \right. \Rightarrow \left[ \begin{array}{l}
\frac{{{R_1}}}{{{R_2}}} = 0,5\\
\frac{{{R_1}}}{{{R_2}}} = 2
\end{array} \right.
\end{array}\)
