We are tasked with sketching the set defined by the equation:

$|z - (3 + 4i)| = |1 - (3 - 4i)z|$

This involves finding the locus of points $z = x + iy$ (where $x$ and $y$ are real) that satisfy the equation. Here’s the step-by-step breakdown of how we would analyze and simplify the equation:

Step 1: Interpret the equation geometrically

The equation involves the absolute values (or magnitudes) of two complex numbers. The magnitude of a complex number $w = a + bi$ is given by $|w| = \sqrt{a^2 + b^2}$.

Thus, the equation can be interpreted geometrically as describing the set of points $z$ in the complex plane where the distance between $z$ and $(3 + 4i)$ equals the distance between $z$ and another expression involving $z$ itself.

Step 2: Analyze the right-hand side

The term $1 - (3 - 4i)z$ is a more complex expression that depends on $z$. To get a clear idea of how this behaves, we can expand it:

$1 - (3 - 4i)z = 1 - (3 - 4i)(x + iy)$

This simplifies as:

$1 - (3x - 4y + i(4x + 3y)) = 1 - 3x + 4y - i(4x + 3y)$

Thus, the magnitude of this term is:

$|1 - (3 - 4i)z| = \sqrt{(1 - 3x + 4y)^2 + (-4x - 3y)^2}$

Step 3: Sketching the set

The equation $|z - (3 + 4i)| = |1 - (3 - 4i)z|$ defines a curve in the complex plane. However, given the complexity of the right-hand side, we won't be able to describe it as a simple geometric figure (such as a line or circle) without further algebraic manipulation. This type of equation generally defines a more intricate locus of points, potentially a curve or boundary depending on the specific relation between $z$ and its transformed values.

To sketch this:

1. $|z - (3 + 4i)|$ describes a circle centered at $(3, 4)$ with a variable radius determined by the right-hand side.
2. The other side describes a more complex curve due to the dependency on $z$.

Conclusion

The set defined by the equation is a complex curve in the plane. A precise sketch would require evaluating the function for various values of $z$ and plotting the points, but qualitatively, the set involves a transformation of the plane due to the $(3 - 4i)z$ term.

Would you like further explanation or help visualizing this graph numerically?

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Here are some related questions to expand on this:

1. What does the transformation by multiplying $z$ by a complex number represent geometrically?
2. How can we solve complex modulus equations algebraically?
3. Can this equation be reduced to a more familiar geometric shape, like a line or circle?
4. What role do the real and imaginary parts of $z$ play in equations like these?
5. How can we interpret transformations like $1 - (3 - 4i)z$ geometrically?

Tip: When working with complex numbers, geometric interpretations often provide deep insights into their properties and transformations.