Complex Numbers
System of Complex Equations
Grade 11
Question:
<p>If <math>z_1, z_2 ∈ ℂ</math>, <math>z_1^2 + z_2^2 ∈ ℝ</math>, <math>z_1(z_1^2 + 3z_2^2) = 2</math> and <math>z_2(3z_1^2 + z_2^2) = 11</math>, find the value of <math>z_1^2 + z_2^2</math>.</p>
Step-by-Step Solution
Key Concept: Use complex number algebraic manipulation by multiplying the equations by $i$ and $-i$ to create cube expressions that can be solved.
<p><strong>Step 1:</strong> We have:</p><p>$$z_1(z_1^2 + 3z_2^2) = 2 \quad \text{...(i)}$$</p><p>$$z_2(3z_1^2 + z_2^2) = 11 \quad \text{...(ii)}$$</p><p><strong>Step 2:</strong> Multiply equation (ii) by $i$ and add to equation (i):</p><p>$$z_1 + 3z_1z_2 + i(3z_1z_2 + z_2^3) = 2 + 11i$$</p><p>$$(z_1 + iz_2)^3 = 2 + 11i \quad \text{...(iii)}$$</p><p><strong>Step 3:</strong> Multiply equation (ii) by $-i$ and add to equation (i):</p><p>$$z_1 + 3z_1z_2 - i(3z_1z_2 + z_2^3) = 2 - 11i$$</p><p>$$(z_1 - iz_2)^3 = 2 - 11i \quad \text{...(iv)}$$</p><p><strong>Step 4:</strong> Multiply equations (iii) and (iv):</p><p>$$(z_1 + iz_2)^3(z_1 - iz_2)^3 = (2 + 11i)(2 - 11i) = 4 + 121 = 125$$</p><p>$$(z_1^2 + z_2^2)^3 = 125$$</p><p>$$z_1^2 + z_2^2 = 5$$</p><p>∴ Answer is <strong>5</strong>.</p>
Correct Answer: 5