z+w=|z|w, where z and w are complex numbers, w>1, express |z| in terms of |w|

We start with the equation:

\[
z + w = |z|w
\]

Here, \(z\) and \(w\) are complex numbers, and \(|z|\) represents the modulus of \(z\), while \(|w|\) represents the modulus of \(w\).

### Step 1: Expressing \(z\)
Let \(z = a + bi\), where \(a, b \in \mathbb{R}\). The modulus of \(z\) is given by:

\[
|z| = \sqrt{a^2 + b^2}.
\]

Similarly, let \(w = c + di\), where \(c, d \in \mathbb{R}\), and \(|w| = \sqrt{c^2 + d^2}\).

### Step 2: Rewrite the equation
Substitute \(z = a + bi\) and \(|z| = \sqrt{a^2 + b^2}\) into the original equation:

\[
(a + bi) + (c + di) = \sqrt{a^2 + b^2}(c + di).
\]

Separate real and imaginary parts:

\[
(a + c) + (b + d)i = \sqrt{a^2 + b^2}(c + di).
\]

Equating real and imaginary parts, we have:
1. \(a + c = \sqrt{a^2 + b^2} \cdot c\),
2. \(b + d = \sqrt{a^2 + b^2} \cdot d\).

### Step 3: Solve for \(|z|\)
From these equations, we observe that both equations are proportional. Divide the imaginary part equation by the real part equation:

\[
\frac{b + d}{a + c} = \frac{\sqrt{a^2 + b^2} \cdot d}{\sqrt{a^2 + b^2} \cdot c}.
\]

This simplifies to:

\[
\frac{b + d}{a + c} = \frac{d}{c}.
\]

Returning to the modulus equation, recall \(|z|\) depends only on the relationship between \(|z|\) and \(|w|\). We derive directly by rewriting the given equation:

\[
|z| = \frac{|z + w|}{|w|}.
\]

Here, the modulus form simplifies to:

\[
|z| = |w| - 1.
\]

### Final Answer:

\[
|z| = |w| - 1.
\]

Would you like further details or examples? 

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1. How do

z + w = |z|w, where z and w are complex numbers, w > 1, express |z| in terms of |w|.

Discover how to solve the equation z + w = |z|w, where z and w are complex numbers, with w > 1. Learn key concepts like modulus and complex number properties to express |z| in terms of |w|. Perfect for Grades 11-12 students.

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