Question

If $\alpha ,\beta$ are roots of the equation $2x^2+x+1=0$ then the equation whose roots are $\frac{1}{\alpha },\frac{1}{\beta }$ will be

• A. $x^2+x+3=0$
• B. $x^2-3x+2=0$
• C. $x^2+x+2=0$
• D. $x^2-x+2=0$

Answer 1

The correct option is C $x^2+x+2=0$

Given: $\alpha ,\beta$ are the roots of equation $2x^2+x-1=0$
$\therefore$ Sum of roots:
$\alpha +\beta =-\frac{1}{2}$
And Product of roots:
$\alpha .\beta =\frac{1}{2}$

Now, if the roots of the equation is $\frac{1}{\alpha },\frac{1}{\beta }$
$\therefore$ We can write the equation as:

$x^2-(\frac{1}{\alpha }+\frac{1}{\beta })x+\frac{1}{\alpha }.\frac{1}{\beta }=0$

$\Rightarrow x^2-\left(\frac{\alpha +\beta }{\alpha .\beta }\right)x+\frac{1}{\alpha .\beta }=0$

$\Rightarrow x^2-\left(\frac{-\frac{1}{2}}{\frac{1}{2}}\right)x+\frac{1}{\frac{1}{2}}=0$

$x^2+x+2=0$