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GATE 2016 EE - SET 2 - Question 11

Using the phasor diagram for this circuit, the following equation can be written. 

V_{S}^{2}=(V_{R}+V_{C}\cos \phi )^{2}+V_{C}^{2}\sin ^{2}\phi

V_{S}^{2}=V_{R}^{2}+V_{C}^{2}\cos ^{2}\phi +2V_{R}V_{C}\cos \phi +V_{C}^{2}\sin ^{2}\phi

V_{S}^{2}=V_{R}^{2}+V_{C}^{2}+2V_{R}V_{C}\cos\phi

\cos\phi=\frac{V_{S}^{2}-V_{R}^{2}-V_{C}^{2}}{2V_{R}V_{C}}

0=\frac{V_{S}^{2}-V_{R}^{2}-V_{C}^{2}}{2V_{R}V_{C}}

Substituting, V_{S}=100 along with the options given into the above equation, we see that only for option D, where V_{R}=60 and V_{C}=80, the equation satisfies. 

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