Question

If \( x + iy = \frac{3 + 4i}{5 - 12i} \), then \( x + y \) is equal to

• A. \( \frac{23}{169} \)
• B. \( \frac{56}{169} \)
• C. \( \frac{15}{169} \)
• D. \( \frac{15}{169} \)
• E. \( \frac{71}{169} \)

Hint:

When dealing with complex fractions, multiply both the numerator and denominator by the conjugate of the denominator.

Answer 1

The Correct Option is A

Step 1: Multiply numerator and denominator by the conjugate of the denominator to simplify: \[ \frac{3 + 4i}{5 - 12i} \cdot \frac{5 + 12i}{5 + 12i} = \frac{(3 + 4i)(5 + 12i)}{(5 - 12i)(5 + 12i)} \] Step 2: Simplify the denominator: \[ (5 - 12i)(5 + 12i) = 5^2 + 12^2 = 25 + 144 = 169 \] Step 3: Simplify the numerator: \[ (3 + 4i)(5 + 12i) = 15 + 36i + 20i + 48i^2 = 15 + 56i - 48 = -33 + 56i \] Step 4: Now, the expression becomes: \[ \frac{-33 + 56i}{169} \] Step 5: This gives \( x = \frac{-33}{169} \) and \( y = \frac{56}{169} \). 
Step 6: Therefore, \( x + y = \frac{-33}{169} + \frac{56}{169} = \frac{23}{169} \).