Complex Numbers Question - None

Question

995. If complex number $z$ satisfies $(z - \bar{z})^2 = 12|z|^2 - 4$ then find the maximum value of $3\sqrt{3}\,\text{Re}(z) + 8\,\text{Im}(z)$.

Accepted Answer

Step 1: Let z = x + iy where x, y ∈ ℝ. Then z̄ = x - iy and |z|² = x² + y².

Step 2: Calculate z - z̄ = 2iy, so (z - z̄)² = (2iy)² = -4y².

Step 3: Substitute into the constraint: -4y² = 12(x² + y²) - 4

-4y² = 12x² + 12y² - 4

-16y² = 12x² - 4

12x² + 16y² = 4

Step 4: Simplify by dividing by 4: 3x² + 4y² = 1

This is an ellipse: $\frac{x^2}{1/3} + \frac{y^2}{1/4} = 1$

Step 5: Parametrize: x = $\frac{1}{\sqrt{3}}\cos heta$, y = $\frac{1}{2}\sin heta$

Step 6: Maximize f = 3√3 Re(z) + 8 Im(z) = 3√3·x + 8y

f = 3√3·$\frac{1}{\sqrt{3}}\cos heta$ + 8·$\frac{1}{2}\sin heta$ = 3cos θ + 4sin θ

Step 7: The maximum of A cos θ + B sin θ is √(A² + B²)

Maximum = √(9 + 16) = √25 = 5

∴ Answer: 5

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