<p>If <i>|z₁|</i> and <i>|z₂|</i> are the distances of points on the curve <i>5zz̄ - 2i(z² - z̄²) - 9 = 0</i> which are at maximum and minimum distance from the origin, then the value of <i>|z₁| + |z₂|</i> is equal to:</p>
Step-by-Step Solution
Key Concept: Convert the complex equation to Cartesian form by substituting z = x + iy, then recognize the resulting curve as a conic section. The distances from the origin to points on this curve are maximized and minimized at the vertices along the principal axis.
<p><strong>Step 1: Substitute z = x + iy and z̄ = x - iy</strong></p><p>We have zz̄ = x² + y²</p><p>z² = (x + iy)² = x² - y² + 2ixy</p><p>z̄² = (x - iy)² = x² - y² - 2ixy</p><p>Therefore: z² - z̄² = 4ixy</p><p><strong>Step 2: Substitute into the given equation</strong></p><p>5(x² + y²) - 2i(4ixy) - 9 = 0</p><p>5(x² + y²) - 8i²xy - 9 = 0</p><p>5(x² + y²) + 8xy - 9 = 0</p><p><strong>Step 3: Identify the curve type</strong></p><p>5x² + 8xy + 5y² - 9 = 0</p><p>This is an ellipse. To find principal axes, diagonalize the quadratic form.</p><p><strong>Step 4: Find eigenvalues of the matrix</strong></p><p>The matrix is A = [[5, 4], [4, 5]]</p><p>Characteristic equation: (5 - λ)² - 16 = 0</p><p>(5 - λ)² = 16</p><p>5 - λ = ±4</p><p>λ₁ = 1, λ₂ = 9</p><p><strong>Step 5: Transform to principal axis form</strong></p><p>The equation becomes: x'² + 9y'² = 9</p><p>Dividing by 9: x'²/9 + y'² = 1</p><p>This is an ellipse with semi-major axis a = 3 and semi-minor axis b = 1</p><p><strong>Step 6: Find maximum and minimum distances</strong></p><p>Maximum distance from origin: |z₁| = 3</p><p>Minimum distance from origin: |z₂| = 1</p><p><strong>Step 7: Calculate the sum</strong></p><p>|z₁| + |z₂| = 3 + 1 = 4</p><p><strong>∴ Answer: 4</strong></p>
Correct Answer: 4