<p>If \(z = (\lambda + 3) - i\sqrt{5 - \lambda^2}\), then the locus of \(z\) is</p>
Step-by-Step Solution
Key Concept: Recognize that z = x + iy where x = λ + 3 and y = -√(5 - λ²). Eliminate the parameter λ by using the constraint that the expression under the square root must be non-negative, which gives a relationship between x and y representing a locus.
<p><strong>Step 1:</strong> Let z = x + iy. Then x = λ + 3 and y = -√(5 - λ²)</p><p><strong>Step 2:</strong> From the imaginary part: y² = 5 - λ², so λ² = 5 - y²</p><p><strong>Step 3:</strong> From the real part: λ = x - 3</p><p><strong>Step 4:</strong> Substitute λ = x - 3 into λ² = 5 - y²:<br/>(x - 3)² = 5 - y²<br/>⟹ (x - 3)² + y² = 5</p><p><strong>Step 5:</strong> Determine the domain: Since -√5 ≤ λ ≤ √5, we have -√5 ≤ x - 3 ≤ √5, giving 3 - √5 ≤ x ≤ 3 + √5</p><p><strong>Step 6:</strong> Also, since y = -√(5 - λ²), we have y ≤ 0 (lower semicircle)</p><p>∴ The locus is the lower semicircle: (x - 3)² + y² = 5 with y ≤ 0</p>
Correct Answer: B