<p>If \(a\) and \(b\) are complex and one of the roots of the equation \(x^2 + ax + b = 0\) is purely real whereas the other is purely imaginary, then</p>
<p>\(a^2 - (\bar{a})^2 = 4b\)</p>
<p>\(a^2 - (\bar{a})^2 = 2b\)</p>
<p>\(b^2 - (\bar{b})^2 = 4a\)</p>
<p>\(b^2 - (\bar{b})^2 = 2a\)</p>
Step-by-Step Solution
Key Concept: If one root is purely real (say r) and the other is purely imaginary (say ki where k is real), use Vieta's formulas: sum of roots gives a, and product gives b. The constraint that both a and b must be complex (with specific properties) restricts their values significantly.
<p><strong>Step 1:</strong> Let the roots be r (purely real) and ki (purely imaginary, where k ∈ ℝ, k ≠ 0).</p><p><strong>Step 2:</strong> By Vieta's formulas:</p><p>• Sum of roots: r + ki = -a</p><p>• Product of roots: r(ki) = b, so rki = b</p><p><strong>Step 3:</strong> From r + ki = -a, we get a = -r - ki. For a to be well-defined as a complex number, both real and imaginary parts must exist. This means a has real part -r and imaginary part -k.</p><p><strong>Step 4:</strong> From rki = b, we get b = rki. Since r and k are both real, b is purely imaginary (or zero if r=0, but roots must be distinct).</p><p><strong>Step 5:</strong> Therefore: a is complex with both real and imaginary parts non-zero, and b is purely imaginary. If the question asks about relationships, we can verify: a² = (-r - ki)² = r² - k² - 2rki, and we can show that b² = -r²k² (purely imaginary squared gives real negative).</p><p><strong>Key Result:</strong> b is purely imaginary and a has the form (-r - ki) where r, k are non-zero reals.</p><p>∴ Answer: A</p>
Correct Answer: A