Question

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

• A. $2$
• B. $4$
• C. $6$
• D. $noneofthese$

Answer 1

Answer: (B)

$4$

$\begin{array}{c}\left(5+\sqrt{2}\right)x^2-\left(4+\sqrt{5}\right)x+8+2\sqrt{5}=0\\ a=5+\sqrt{2}\\ b=-\left(4+\sqrt{5}\right)\\ c=8+2\sqrt{5}\\ Harmonicmeanof\lambda ,\beta \\ =\frac{2\lambda \beta }{\lambda +\beta }\\ \lambda \beta =\frac{c}{a}=\frac{8+2\sqrt{5}}{5+\sqrt{2}}\\ \lambda +\beta =\frac{-b}{a}=\frac{4+\sqrt{5}}{5+\sqrt{2}}\\ Harmonicmean=\\ \frac{2\frac{\left(8+2\sqrt{5}\right)}{5\sqrt{2}}}{\frac{4+\sqrt{5}}{5+\sqrt{2}}}\\ =\frac{2\left(8+2\sqrt{5}\right)}{4+\sqrt{5}}\\ =4\end{array}$