Question

If $\alpha$ and $\beta$ are the roots of $a{x}^2+bx+c=0$ find the equation having roots $\frac{1}{\alpha }$ and $\frac{1}{\beta }$.

• A. $b{x}^2+cx+a=0$
• B. $c{x}^2+ax+b=0$
• C. $c{x}^2+bx+a=0$
• D. $c{x}^2-bx+a=0$

Answer 1

Answer: (C)

$c{x}^2+bx+a=0$

$\alpha$ and $\beta$ are the roots of $a{x}^2+bx+c=0$

So,$\alpha +\beta =\frac{-b}{a}$

and $\alpha \beta =\frac{c}{a}$

Let $\frac{1}{\alpha }$ and $\frac{1}{\beta }$ be the roots of new polynomial $g\left(x\right)=0$

So, sum of roots $=\frac{1}{\alpha }+\frac{1}{\beta }=\frac{\alpha +\beta }{\alpha \beta }=\frac{\frac{-b}{a}}{\frac{c}{a}}=\frac{-b}{c}$

and product of roots $\frac{1}{\alpha \beta }=\frac{a}{c}$

So, $g\left(x\right)=x^2-$(sum of roots)$x+$(product of roots)

So, $g\left(x\right)=x^2-\left(\frac{-b}{c}\right)x+\frac{a}{c}$

So, $g\left(x\right)=x^2+\frac{b}{c}x+\frac{a}{c}$

So, the required equation is $x^2+\frac{b}{c}x+\frac{a}{c}=0$

$\therefore c{x}^2+bx+a=0$

The answer is option (C)