Question

If $\alpha$ and $\beta$ are the roots of the equation $5x^2-x-2=0,$ then the equation for which roots are $\frac{2}{\alpha }$ and $\frac{2}{\beta }$ is

• A. $x^2-x+10=0$
• B. $x^2-x-10=0$
• C. $x^2+x+10=0$
• D. $x^2+x-10=0$

Answer 1

Answer: (D)

$x^2+x-10=0$

Since $\alpha ,\beta$ are roots of the equation

$5x^2-x-2=0$

$\therefore \alpha +\beta =\frac{1}{5}$ and $\therefore \alpha \beta =\frac{-2}{5}$

Now sum of roots of required equation

$=\frac{2}{\alpha }+\frac{2}{\beta }=\frac{2(\alpha +\beta )}{\alpha \beta }=\frac{2\left(\frac{1}{5}\right)}{\left(\frac{-2}{5}\right)}=-1$

and product of roots of required equation

$=\frac{2}{\alpha }.\frac{2}{\beta }=\frac{4}{\alpha \beta }=\frac{4}{\frac{-2}{5}}=-10$

Hence required equation is

$x^2-(-1)x+(-10)=0$

or $x^2+x-10=0$