Question

If A.M. of the roots of a quadratic equation is $8/5$ and A.M. of their reciprocals is $8/7$, then the equation is?

• A. $7x^2-16x+8=0$
• B. $3x^2-12x+7=0$
• C. $5x^2-16x+7=0$
• D. $7x^2-16x+5=0$

Answer 1

The correct option is C $5x^2-16x+7=0$

$\Rightarrow$ Let $\alpha$ and $\beta$ are the roots of the equation.

According to the question,

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

$\therefore$ $\alpha +\beta =\frac{16}{5}$ ----- ( 1 )

According to the equation,

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

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

$\Rightarrow$ $7(\alpha +\beta )=16\alpha \beta$

$\Rightarrow$ $\frac{7\times \frac{16}{5}}{16}=\alpha \beta$

$\therefore$ $\alpha \beta =\frac{7}{5}$ - ------ ( 2 )

Now, new eqution,

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

From ( 1 ) and ( 2 ),

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

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