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.