Question

For real numbers \( a \) and \( b \), if \[ 4a + i(3a - b) = b - 6i \] and \[ z = a + \frac{b}{4}i, \] then find \[ \frac{|z|}{a}. \]

• A. \( 2\sqrt{2} \)
• B. \( 6\sqrt{2} \)
• C. \( \sqrt{2} \)
• D. \( 2 \)

Hint:

For modulus of a complex number \( a + bi \), use the formula \( |a + bi| = \sqrt{a^2 + b^2} \), and always match real and imaginary parts when equating complex numbers.

Answer 1

The Correct Option is C

From the given: \[ 4a + i(3a - b) = b - 6i \] Equating real and imaginary parts: \[ 4a = b \] and \[ 3a - b = -6 \] Substituting \( b = 4a \) into second: \[ 3a - 4a = -6 \Rightarrow -a = -6 \Rightarrow a = 6 \] Then, \( b = 4a = 24 \) Now, \[ z = 6 + \frac{24}{4}i = 6 + 6i \] Then, \[ |z| = \sqrt{6^2 + 6^2} = \sqrt{72} = 6\sqrt{2} \] Finally, \[ \frac{|z|}{a} = \frac{6\sqrt{2}}{6} = \sqrt{2} \]