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Q.
The equivalent resistance of two resistors connected in series is $6\, \Omega$ and their parallel equivalent resistance is $\frac{4}{3} \Omega$. What are the values of resistances ?
Let the value of resistance be $R_{1}$ and $R_{2}$ respectively.
When $R_{1}$ and $R_{2}$ resistances are in series
So, $R_{1}+R_{2}=6 \Omega\,\,\,\,\,\dots(i)$
(According to the question)
When $R_{1}$ and $R_{2}$ resistances are in parallel
So, $\frac{R_{1} R_{2}}{R_{1}+R_{2}}=\frac{4}{3} \Omega\,\,\,\,\, \dots(ii)$
From the Eq. (i), we get
$\frac{R_{1} R_{2}}{6}=\frac{4}{3}$
$R_{1} R_{2}=4 \times 2$
$R_{1} R_{2}=8\,\,\,\,\,\dots(iii)$
We know that
$R_{1}-R_{2} =\sqrt{\left(R_{1}+R_{2}\right)^{2}-4 R_{1} R_{2}}$
$=\sqrt{36-4 \times 8}$
$R_{1}-R_{2} =\sqrt{4}$
$R_{1}-R_{2} =2 \Omega\,\,\,\,\,\dots(iv)$
From the Eqs. (i) and (iv), we get
$R_{1}=4 \Omega, R_{2}=2 \Omega$