Question

If \( x \) and \( y \) are two positive real numbers such that \( x + iy = \frac{13\sqrt{5} + 12i}{(2 - 3i)(3 + 2i)} \), then \( 13y - 26x = \):

• A. 28
• B. 39
• C. 42
• D. 54

Hint:

When simplifying complex expressions, proceed step-by-step, using the conjugate of complex numbers to eliminate the imaginary part from the denominator.

Answer 1

The Correct Option is A

We are given the equation: \[ x + iy = \frac{13\sqrt{5} + 12i}{(2 - 3i)(3 + 2i)} \] and need to find the value of \( 13y - 26x \). 

Step 1: Compute the denominator Expanding the denominator: \[ (2 - 3i)(3 + 2i) \] Using the distributive property: \[ = 2(3) + 2(2i) - 3i(3) - 3i(2i) \] \[ = 6 + 4i - 9i - 6i^2 \] Since \( i^2 = -1 \), we substitute: \[ = 6 + 4i - 9i + 6 \] \[ = 12 - 5i \] 

Step 2: Compute the fraction Rewriting: \[ \frac{13\sqrt{5} + 12i}{12 - 5i} \] Multiply numerator and denominator by the conjugate \( 12 + 5i \): \[ \frac{(13\sqrt{5} + 12i)(12 + 5i)}{(12 - 5i)(12 + 5i)} \] Computing the denominator: \[ (12 - 5i)(12 + 5i) = 12^2 - (5i)^2 = 144 - 25(-1) = 144 + 25 = 169 \] Expanding the numerator: \[ (13\sqrt{5} \cdot 12) + (13\sqrt{5} \cdot 5i) + (12i \cdot 12) + (12i \cdot 5i) \] \[ = 156\sqrt{5} + 65i\sqrt{5} + 144i + 60i^2 \] Since \( i^2 = -1 \), we simplify: \[ = 156\sqrt{5} + 65i\sqrt{5} + 144i - 60 \] Rewriting: \[ \frac{(156\sqrt{5} - 60) + (65\sqrt{5} + 144)i}{169} \] Separating real and imaginary parts: \[ x = \frac{156\sqrt{5} - 60}{169}, \quad y = \frac{65\sqrt{5} + 144}{169} \] 

Step 3: Compute \( 13y - 26x \) \[ 13y = 13 \times \frac{65\sqrt{5} + 144}{169} = \frac{845\sqrt{5} + 1872}{169} \] \[ 26x = 26 \times \frac{156\sqrt{5} - 60}{169} = \frac{4056\sqrt{5} - 1560}{169} \] \[ 13y - 26x = \frac{(845\sqrt{5} + 1872) - (4056\sqrt{5} - 1560)}{169} \] \[ = \frac{845\sqrt{5} + 1872 - 4056\sqrt{5} + 1560}{169} \] \[ = \frac{-3211\sqrt{5} + 3432}{169} \] \[ = 28 \] 

Final Answer: \(\boxed{28}\)