In the given figure if points B and D are at same potential then the…

The correct answer is Option (2) → \( 3\,\Omega \)

If B and D are at the same potential, the drops \( A\to B \) and \( A\to D \) are equal, and so are the drops \( B\to C \) and \( D\to C \). This is the balanced Wheatstone bridge condition, and no current flows between B and D:

    \( \displaystyle \frac{R_{AB}}{R_{BC}} = \frac{R_{AD}}{R_{DC}} \)

Readings from the figure

    \( R_{AB} \): \( 6\,\Omega \) in parallel with x, so \( \displaystyle R_{AB} = \frac{6x}{6+x} \)
    \( R_{BC} = 6\,\Omega \)
    \( R_{AD} = 1\,\Omega + 4\,\Omega = 5\,\Omega \) (series)

    \( R_{DC} = 10\,\Omega + (10\,\Omega\parallel10\,\Omega) = 10\,\Omega + 5\,\Omega = 15\,\Omega \)

Balance

    \( \displaystyle \frac{6x/(6+x)}{6} = \frac{5}{15} = \frac{1}{3} \Rightarrow \frac{6x}{6+x} = 2 \)

    \( 6x = 12 + 2x \Rightarrow x = 3\,\Omega \)