Question

If the arithmetic mean of the roots of a quadratic equation is $\frac{8}{5}$ and the arithmetic mean of their reciprocals is $\frac{8}{7}$, then the equation is

• A. $5x^2+16x+7=0$
• B. $5x^2-16x+7=0$
• C. $7x^2+16x+5=0$
• D. $7x^2-16x+5=0$

Answer 1

Answer: (B)

$5x^2-16x+7=0$

Let the roots be $\alpha ,\beta$

Hence

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

$\alpha +\beta =\frac{16}{5}$ ...(i)

And

$\frac{\frac{1}{\alpha }+\frac{1}{\beta }}{2}=\frac{8}{7}$

$\frac{\alpha +\beta }{\alpha .\beta }=\frac{16}{7}$

$\alpha +\beta =\frac{16}{7}(\alpha .\beta )$

$\frac{16}{5}=\frac{16}{7}(\alpha .\beta )$ ...from i

$\alpha .\beta =\frac{7}{5}$ ...(ii)

Thus the equation is

$(x-\alpha )(x-\beta )=0$
$x^2-(\alpha +\beta )x+\alpha .\beta =0$

$x^2-\left(\frac{16}{5}\right)x+\frac{7}{5}=0$

$5x^2-16x+7=0$