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.

sketch the set { |z-(3+4i)|=|1-(3-4i)z| }

Explore how to sketch the set of complex numbers defined by the equation |z - (3 + 4i)| = |1 - (3 - 4i)z|. This problem covers key concepts like the modulus of complex numbers, geometric interpretation of loci, and transformations in the complex plane. Suitable for advanced high school or undergraduate students.

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