To explore how the given transformed equation represents a line, let’s go through the transformation in detail:

1. Starting with the Circle Equation:
   The equation of a circle in the complex plane is given by: $|z - c| = r$ where $z$ is a complex variable, $c$ is the fixed center of the circle (a complex constant), and $r$ is the radius of the circle, a positive real constant.

2. Transforming with the Inversion $u = \frac{1}{z}$:
   We are given the substitution $u = \frac{1}{z}$, which implies $z = \frac{1}{u}$ whenever $u \neq 0$. This is a Möbius transformation (specifically an inversion) in the complex plane. Under this transformation, circles in the complex plane that do not pass through the origin can be mapped to other circles, and circles that do pass through the origin are mapped to lines.

3. Substitute $z = \frac{1}{u}$ into the Circle Equation:
   Substitute $z = \frac{1}{u}$ into $|z - c| = r$: $\left| \frac{1}{u} - c \right| = r$ This simplifies to: $\left| \frac{1 - cu}{u} \right| = r$ or, equivalently: $|1 - cu| = r |u|$

4. Interpretation of the Transformed Equation:
   Now we have the equation: $|1 - cu| = r |u|$ To understand this as a line in the complex plane, rewrite it as: $\left| \frac{1}{u} - c \right| = r$ which represents points $u$ for which the modulus $|1 - cu| = r |u|$ holds. This equation, under certain conditions on $r$, $c$, and $u$, represents a locus of points that forms a line in the complex $u$-plane. This happens specifically because the Möbius transformation $u = \frac{1}{z}$ maps the circle passing through the origin in the $z$-plane to a line in the $u$-plane.

In summary, the Möbius transformation $u = \frac{1}{z}$ maps the original circle in the $z$-plane to a line in the $u$-plane because it inverts the geometry of the circle around the origin, transforming circular shapes that pass through the origin into linear forms in the transformed plane.

Would you like more details on how Möbius transformations specifically map circles to lines?

Here are five related questions:

1. How does a Möbius transformation generally affect shapes in the complex plane?
2. Can all circles be mapped to lines under some Möbius transformation?
3. What are the fixed points of the transformation $u = \frac{1}{z}$?
4. How can the general form of a Möbius transformation be expressed?
5. What happens when applying $u = \frac{1}{z}$ to lines in the complex plane?

Tip: Möbius transformations can map the extended complex plane (including infinity) into itself, making them very powerful for geometric transformations in complex analysis.