Complex Numbers
Basic Complex Number Operations
Grade 11
Question:
<p>For all complex numbers <em>z</em> of the form \(1 + i\alpha,\ \alpha \in R\), if \(z^2 = x + iy\), then</p>
<p>\(y^2 - 4x + 2 = 0\)</p>
<p>\(y^2 + 4x - 4 = 0\)</p>
<p>\(y^2 - 4x + 4 = 0\)</p>
<p>\(y^2 + 4x + 2 = 0\)</p>
Step-by-Step Solution
Key Concept: When z = 1 + iα is squared, the real and imaginary parts of z² = (1 + iα)² reveal a relationship between x and y that is independent of the parameter α. This relationship is the locus equation connecting x and y.
<p><strong>Step 1:</strong> Expand z² where z = 1 + iα</p><p>z² = (1 + iα)² = 1 + 2iα + (iα)² = 1 + 2iα - α²</p><p>z² = (1 - α²) + i(2α)</p><p><strong>Step 2:</strong> Identify real and imaginary parts</p><p>x = 1 - α² and y = 2α</p><p><strong>Step 3:</strong> Eliminate the parameter α</p><p>From y = 2α, we get α = y/2</p><p>Substitute into x = 1 - α²:</p><p>x = 1 - (y/2)² = 1 - y²/4</p><p><strong>Step 4:</strong> Rearrange to standard form</p><p>x + y²/4 = 1, or equivalently: y² = 4(1 - x)</p><p>This represents a parabola with vertex at (1, 0) opening to the left.</p><p>∴ The locus is y² + 4x = 4 (or y² = 4(1 - x))</p>
Correct Answer: C