Question

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

• A. $5x^2-8x+7=0$
• 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 $\alpha ,\beta$ be the roots of the required equation.
Given,
$\frac{\alpha +\beta }{2}=\frac{8}{5}$
$\Rightarrow \alpha +\beta =\frac{16}{5}$

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

$\frac{1}{\alpha }+\frac{1}{\beta }=\frac{16}{7}$

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

$\Rightarrow \alpha \beta =\frac{7}{5}$
So, the quadratic equation is
$x^2-(\alpha +\beta )+\alpha \beta =0$

$\Rightarrow x^2-\frac{16}{5}x+\frac{7}{5}=0$

Multiply by $5$

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