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.