Question

If the harmonic means between the roots of $(5+\sqrt{2})x^2-bx++(8+2\sqrt{5})=0$ is $4$, then the value of $b$

• A. $2$
• B. $3$
• C. $4-\sqrt{5}$
• D. $4+\sqrt{5}$

Answer 1

The correct option is D $4+\sqrt{5}$

$(5+\sqrt{2})x^2-bx+(8+2\sqrt{5})=0$

Let the roots $\alpha ,\beta$

$\alpha +\beta =\frac{b}{5+\sqrt{5}}$

$\alpha \beta =\frac{8+2\sqrt{5}}{5+\sqrt{5}}$

Harmonic mean $=\frac{2\alpha \beta }{\alpha +\beta }=\frac{2(8+2\sqrt{5})}{b}$

$\Rightarrow 4=\frac{2(8+2\sqrt{5})}{b}$

$\Rightarrow 2b=8+2\sqrt{5}$

$\Rightarrow b=4+\sqrt{5}$