Let $S$ be the set of complex numbers $z$ such that the real part of $\frac{1}{z}$ is equal to $\frac{1}{6}.$  This set forms a curve.  Find the area of the region inside the curve.
Question:
Let $S$ be the set of complex numbers $z$ such that the real part of $\frac{1}{z}$ is equal to $\frac{1}{6}.$  This set forms a curve.  Find the area of the region inside the curve.
Answer:
Let's think step by step.

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9 \pi
Let $S$ be the set of complex numbers $z$ such that the real part of $\frac{1}{z}$ is equal to $\frac{1}{6}.$  This set forms a curve.  Find the area of the region inside the curve.
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