Question

Find the quadratic equation whose roots are half of the reciprocal of the roots of the equation $a{x}^2+bx+c=0$

• A. $4a{x}^2+2bx+c=0$
• B. $4c{x}^2+2bx+a=0$
• C. $2c{x}^2+bx+a=0$
• D. $2a{x}^2+bx+c=0$

Answer 1

The correct option is B $4c{x}^2+2bx+a=0$

Let $\alpha ,\beta$ be the roots of the equation $a{x}^2+bx+c=0$

The sum of the roots are $\alpha +\beta =-b/a$ and the product of the roots are $\alpha \beta =c/a$.

The new roots would be ${\alpha }^{\prime }=1/2\alpha and{\beta }^{\prime }=1/2\beta$.

The new equation formed will be $x^2-\left(sumofroots\right)x+\left(productofroots\right)=0$

${\alpha }^{\prime }+{\beta }^{\prime }=\left(1/2\alpha \right)+\left(1/2\beta \right)=(\alpha +\beta )/\left(2\alpha \beta \right)$

${\alpha }^{\prime }{\beta }^{\prime }=1/4\alpha \beta$

$x^2-\left(\frac{\alpha +\beta }{2\alpha \beta }\right)x+\left(1/4\alpha \beta \right)=04\alpha \beta x^2-2(\alpha +\beta )x+1=04\frac{c}{a}{x}^2-2\frac{-b}{a}x+1=04c{x}^2+2bx+a=0$

Hence option B.