Question

By taking $ \sqrt{a \pm ib} = x + iy, x>0 $, if we get $$ \frac{\sqrt{21} + 12\sqrt{2}i}{\sqrt{21} - 12\sqrt{2}i} = a + ib, $$ then $ \frac{b}{a} = $ ?

• A. \( \frac{4\sqrt{2}}{7} \)
• B. \( \frac{12\sqrt{2}}{17} \)
• C. \( \frac{4\sqrt{3}}{7} \)
• D. \( \frac{12\sqrt{3}}{17} \)

Hint:

Always multiply by the conjugate to rationalize complex fractions and extract real and imaginary parts for ratios.

Answer 1

The Correct Option is A

Let us denote the given expression as: \[ \frac{\sqrt{21} + 12\sqrt{2}i}{\sqrt{21} - 12\sqrt{2}i} \] Multiply numerator and denominator by the conjugate of the denominator: \[ = \frac{(\sqrt{21} + 12\sqrt{2}i)(\sqrt{21} + 12\sqrt{2}i)}{(\sqrt{21})^2 + (12\sqrt{2})^2} = \frac{a + ib}{a} \Rightarrow \text{Evaluate } \frac{b}{a} \] After simplification, you get \( \frac{b}{a} = \frac{4\sqrt{2}}{7} \)