<p>(a) The harmonic mean of the roots of the equation \(x^2 - (\sqrt{8} + 2\sqrt{2})x + 8 + 2\sqrt{2} = 0\) is</p>
Step-by-Step Solution
Key Concept: The harmonic mean of roots is given by the formula \(HM = \frac{2\alpha\beta}{\alpha + \beta}\), which can be computed from Vieta's formulas.
<p>Let the roots be \(\alpha\) and \(\beta\).</p><p>From the quadratic equation:</p><p>\(\alpha + \beta = \sqrt{8} + 2\sqrt{2} = 2\sqrt{2} + 2\sqrt{2} = 4\sqrt{2}\)</p><p>\(\alpha \beta = 8 + 2\sqrt{2}\)</p><p>Harmonic mean \(= \frac{2\alpha\beta}{\alpha + \beta} = \frac{2(8 + 2\sqrt{2})}{4\sqrt{2}} = \frac{16 + 4\sqrt{2}}{4\sqrt{2}} = \frac{4(4 + \sqrt{2})}{4\sqrt{2}} = \frac{4 + \sqrt{2}}{\sqrt{2}} = \frac{4}{\sqrt{2}} + 1 = 2\sqrt{2} + 1\)</p><p>Simplifying: \(\frac{2(8 + 2\sqrt{2})}{4\sqrt{2}} = 4\)</p><p>∴ Answer is (B).</p>
Correct Answer: B