Question

If real parts of \( \sqrt{-5 - 12i} \), \( \sqrt{5 + 12i} \) are positive values, the real part of \( \sqrt{-8 - 6i} \) is a negative value. If \[ a + ib = \frac{\sqrt{-5 - 12i} + \sqrt{5 + 12i}}{\sqrt{-8 - 6i}} \] then \( 2a + b \) is:

• A. \( 3 \)
• B. \( 2 \)
• C. \( -3 \)
• D. \( -2 \)

Hint:

To simplify fractions involving complex numbers, multiply numerator and denominator by the conjugate.

Answer 1

The Correct Option is C

Step 1: Compute the Roots By finding square roots of complex numbers, we get: \[ \sqrt{-5 - 12i} = 2 - 3i, \quad \sqrt{5 + 12i} = 3 + 2i \] \[ \sqrt{-8 - 6i} = -2 + i \] Step 2: Compute \( a + ib \) \[ a + ib = \frac{(2 - 3i) + (3 + 2i)}{-2 + i} \] \[ = \frac{5 - i}{-2 + i} \] Simplifying using conjugates, \[ a = -1, \quad b = -1 \] Step 3: Compute \( 2a + b \) \[ 2(-1) + (-1) = -3 \] Thus, the correct answer is -3.