Suppose  z1  and  z2  are complex numbers with  z1z¯2=2+4i . Find  |z1z2|

Given that \( z_1 \) and \( z_2 \) are complex numbers and \( z_1 \overline{z_2} = 2 + 4i \), we want to find the modulus \( |z_1 z_2| \).

### Step 1: Use the property of moduli of complex numbers
For any two complex numbers \( z_1 \) and \( z_2 \), the modulus of their product satisfies:
\[
|z_1 z_2| = |z_1| \cdot |z_2|
\]
We are given that \( z_1 \overline{z_2} = 2 + 4i \). The modulus of this expression can be found as:
\[
|z_1 \overline{z_2}| = |z_1| \cdot |\overline{z_2}|
\]
But since the modulus of a complex number and its conjugate are equal, \( |\overline{z_2}| = |z_2| \). Therefore:
\[
|z_1 \overline{z_2}| = |z_1| \cdot |z_2|
\]
So, \( |z_1 z_2| = |z_1 \overline{z_2}| \).

### Step 2: Find the modulus of \( z_1 \overline{z_2} \)
Given \( z_1 \overline{z_2} = 2 + 4i \), the modulus of this complex number is:
\[
|2 + 4i| = \sqrt{2^2 + 4^2} = \sqrt{4 + 16} = \sqrt{20} = 2\sqrt{5}
\]

### Final Answer
Thus, \( |z_1 z_2| = 2\sqrt{5} \).

Would you like a more detailed explanation of any step? Here are some related questions you might find interesting:

1. How do you find the modulus of a complex number?
2. What is the significance of the conjugate in complex number operations?
3. Can you solve for \( |z_1| \) and \( |z_2| \) individually given more information?
4. How does the product of moduli relate to the moduli of products in complex numbers?
5. What are some applications of complex number operations in physics?

**Tip:** Always remember that the modulus of a complex number is the distance from the origin to the point representing the number in the complex plane.

Suppose z1 and z2 are complex numbers with z1z¯2=2+4i. Find |z1z2|.

Explore how to find the modulus |z1z2| given the complex numbers z1 and z2, where z1z¯2 = 2 + 4i. This solution involves key concepts in complex numbers, including modulus and conjugates. Ideal for undergraduate students studying complex analysis.

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