Question

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

• A. 2
• B. 4
• C. 6
• D. 8

Answer 1

Answer: (B)

4

Answer 2

Let $\alpha, \beta$ be the roots of given quadratic equation. then,
\hspace30mm \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
\hspace30mm H=\frac{2\, \alpha\beta}{\alpha+\beta} =\frac{16 + 4\sqrt 5}{4 + \sqrt 5}=4