Question

If $\alpha$ and $\beta$ are the roots of $a{x}^2+bx+c=0$, then the quadratic equation whose roots are $\frac{1}{\alpha }$ and $\frac{1}{\beta }$ is

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

Answer 1

Answer: (C)

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

The quadratic equation whose roots are $\frac{1}{\alpha }$ & $\frac{1}{\beta }$ is $x^2-(\frac{1}{\alpha }+\frac{1}{\beta })x+\frac{1}{\alpha \beta }$

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

also we know that $\alpha +\beta =\frac{-b}{a}$ and $\alpha \beta =\frac{c}{a}$ as $\alpha ,\beta$ are roots of the equation $a{x}^2+bx+c$

=> $x^2-\left(\frac{-b}{c}\right)x+\frac{a}{c}=0$

=> $x^2+\left(\frac{b}{c}\right)x+\frac{a}{c}=0$

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