The modulus of the complex number `((2 + isqrt(5)))/(1 + 2sqrt(2i))` is :

Options

A. `-1`
B. `1`
C. `sqrt(3)`
D. `-sqrt(3)`

Answer

To find the modulus of the complex number \(\frac{2 + i\sqrt{5}}{1 + 2\sqrt{2}i}\), we can follow these steps:

### Step 1: Identify the complex numbers
Let \( z_1 = 2 + i\sqrt{5} \) and \( z_2 = 1 + 2\sqrt{2}i \).

### Step 2: Calculate the modulus of \( z_1 \)
The modulus of a complex number \( z = x + iy \) is given by:
\[ |z| = \sqrt{x^2 + y^2} \]
For \( z_1 = 2 + i\sqrt{5} \):
- \( x = 2 \)
- \( y = \sqrt{5} \)

Now, calculate the modulus:
\[ |z_1| = \sqrt{2^2 + (\sqrt{5})^2} = \sqrt{4 + 5} = \sqrt{9} = 3 \]

### Step 3: Calculate the modulus of \( z_2 \)
For \( z_2 = 1 + 2\sqrt{2}i \):
- \( x = 1 \)
- \( y = 2\sqrt{2} \)

Now, calculate the modulus:
\[ |z_2| = \sqrt{1^2 + (2\sqrt{2})^2} = \sqrt{1 + 8} = \sqrt{9} = 3 \]

### Step 4: Calculate the modulus of the fraction
The modulus of the fraction of two complex numbers is given by:
\[ \left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|} \]
Substituting the values we found:
\[ \left|\frac{z_1}{z_2}\right| = \frac{3}{3} = 1 \]

### Final Answer
The modulus of the complex number \(\frac{2 + i\sqrt{5}}{1 + 2\sqrt{2}i}\) is \(1\).

---