Quadratic Equations
Nature of roots
Grade 11
Question:
<p>If \(0 < a < b < c\) and the roots \(\alpha, \beta\) of the equation \(ax^2 + bx + c = 0\) are imaginary, then</p>
<p>(a) \(|\alpha| = |\beta|\)</p>
<p>(b) \(|\alpha| > 1\)</p>
<p>(c) \(|\beta| < 1\)</p>
<p>(d) \(|\alpha| = 1\)</p>
Step-by-Step Solution
Key Concept: For imaginary roots of a quadratic, they are complex conjugates with equal magnitudes.
<p>For imaginary roots with \(a, b, c > 0\), the discriminant \(b^2 - 4ac < 0\). By Vieta's formulas, the product of roots \(\alpha\beta = c/a > 1\). Since roots are complex conjugates, \(|\alpha| = |\beta|\) and both satisfy \(|\alpha|^2 = c/a > 1\), so \(|\alpha| > 1\).</p>
Correct Answer: A, B