Find the modulus of the following
`(4 +3i)/(5-12i)`

Answer

To find the modulus of the complex number \((4 + 3i)/(5 - 12i)\), we can use the property of moduli of complex numbers. The modulus of a quotient of two complex numbers can be expressed as the quotient of their moduli.

### Step-by-Step Solution:

1. **Identify the Complex Numbers**:
Let \( z_1 = 4 + 3i \) and \( z_2 = 5 - 12i \).

2. **Apply the Modulus Property**:
The modulus of the quotient is given by:
\[ \left| \frac{z_1}{z_2} \right| = \frac{|z_1|}{|z_2|} \]

3. **Calculate the Modulus of \( z_1 \)**:
The modulus of a complex number \( z = x + yi \) is given by:
\[ |z| = \sqrt{x^2 + y^2} \]
For \( z_1 = 4 + 3i \):
- Here, \( x = 4 \) and \( y = 3 \).
- Therefore,
\[ |z_1| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

4. **Calculate the Modulus of \( z_2 \)**:
For \( z_2 = 5 - 12i \):
- Here, \( x = 5 \) and \( y = -12 \).
- Therefore,
\[ |z_2| = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]

5. **Calculate the Modulus of the Quotient**:
Now, substituting back into our modulus property:
\[ \left| \frac{z_1}{z_2} \right| = \frac{|z_1|}{|z_2|} = \frac{5}{13} \]

### Final Answer:
The modulus of the complex number \(\frac{4 + 3i}{5 - 12i}\) is \(\frac{5}{13}\).

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