Find the modulus of ` (2 -3i)/( 4+i)`

Answer

To find the modulus of the complex number \( \frac{2 - 3i}{4 + i} \), we can follow these steps:

### Step 1: Use the property of modulus
We know that the modulus of a quotient of complex numbers can be expressed as the quotient of their moduli:
\[ \left| \frac{z_1}{z_2} \right| = \frac{|z_1|}{|z_2|} \]
In our case, \( z_1 = 2 - 3i \) and \( z_2 = 4 + i \). Therefore, we can write:
\[ \left| \frac{2 - 3i}{4 + i} \right| = \frac{|2 - 3i|}{|4 + i|} \]

### Step 2: Calculate the modulus of \( z_1 = 2 - 3i \)
The modulus of a complex number \( a + bi \) is given by:
\[ |a + bi| = \sqrt{a^2 + b^2} \]
For \( z_1 = 2 - 3i \):
- \( a = 2 \)
- \( b = -3 \)

Calculating the modulus:
\[ |2 - 3i| = \sqrt{2^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13} \]

### Step 3: Calculate the modulus of \( z_2 = 4 + i \)
For \( z_2 = 4 + i \):
- \( a = 4 \)
- \( b = 1 \)

Calculating the modulus:
\[ |4 + i| = \sqrt{4^2 + 1^2} = \sqrt{16 + 1} = \sqrt{17} \]

### Step 4: Substitute back into the modulus expression
Now substituting the moduli back into our expression:
\[ \left| \frac{2 - 3i}{4 + i} \right| = \frac{|2 - 3i|}{|4 + i|} = \frac{\sqrt{13}}{\sqrt{17}} \]

### Step 5: Final answer
Thus, the modulus of \( \frac{2 - 3i}{4 + i} \) is:
\[ \frac{\sqrt{13}}{\sqrt{17}} \]