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  4. The harmonic mean of the roots of the equation (5+√ 2) x2 - (4+√ 5) x + 8 + 2√5 = 0 is

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Q. The harmonic mean of the roots of the equation $ (5+\sqrt 2)\, x^2 - (4+\sqrt 5)\, x + 8 + 2\sqrt5 = 0 $ is

IIT JEEIIT JEE 1999Sequences and Series

Solution:

Let $ \alpha, \beta $ be the roots of given quadratic equation. then,
$ \alpha+\beta =\frac{4 +\sqrt 5}{5 + \sqrt 2}$ and $ \alpha\beta =\frac{8 + 2\sqrt 5}{5 + \sqrt 2}$
Let H be the harmonic mean between $ \alpha $ and $ \beta $, then
$ H=\frac{2\, \alpha\beta}{\alpha+\beta} =\frac{16 + 4\sqrt 5}{4 + \sqrt 5}=4$