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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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