For the same power to be dissipated in each of the three resistors R_…

The correct answer is Option (3) → \( R_2=R_3 \) and \( R_1=\frac{R_2}{4} \)

From the figure, \( R_2 \) and \( R_3 \) are in parallel, so they have the same potential difference \( V_p \). \( R_1 \) is in series with this pair, so it carries the whole current \( I \). The power in a resistor is \( V^{2}/R \) or \( I^{2}R \).

Equal power in \( R_2 \) and \( R_3 \), with the same \( V_p \), gives:

    \( \displaystyle \frac{V_p^{2}}{R_2}=\frac{V_p^{2}}{R_3} \)

    \( \Rightarrow\; R_2=R_3 \)

So the current \( I \) divides equally. Each of \( R_2 \) and \( R_3 \) carries \( I/2 \). Equal power in \( R_1 \) and \( R_2 \) gives:

    \( \displaystyle I^{2}R_1=\left(\frac{I}{2}\right)^{2}R_2 \)

    \( \displaystyle \Rightarrow\; R_1=\frac{R_2}{4} \)