Figure shows a circuit model for the transmission of an electrical signal, such as cable TV, to a large number of subscribers. Each subscriber connects a load resistance R_L between the transmission line and the ground. Assume the ground to be at zero potential and to have negligible resistance. The resistance of the transmission line between the connection points of different subscribers is modeled as the constant resistance R_T. The equivalent resistance across the signal source is

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$10 Ω$

$\sqrt{55} Ω$

$5 Ω$

$\sqrt{65} Ω$

answer is B.

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Detailed Solution

From the hint, the equivalent resistance of the circuit is

$R_{T} + \frac{1}{1 / R_{L} + 1 / R_{eq}} = R_{eq}$

$\begin{array}{l} R_{T} + \frac{R_{L} R_{eq}}{R_{L} + R_{eq}} = R_{eq} \\ or R_{T} R_{L} + R_{T} R_{eq} + R_{L} R_{eq} = R_{L} R_{eq} + R_{eq}^{2} \\ or R_{eq}^{2} - R_{T} R_{eq} - R_{T} R_{L} = 0 \\ or R_{eq} = \frac{R \pm \sqrt{R_{T}^{2} 4 (1) (- R_{T} R_{L})}}{2 (1)} \end{array}$