Question

If $\alpha ,\beta$ are the roots of the equation $a{x}^2+bx+c=0$ and the equation having roots $\frac{1-\alpha }{\alpha }$ and $\frac{1-\beta }{\beta }$ is $p{x}^2+qx+r=0$, then $r=$

• A. $a+2b$
• B. $ab+bc+ca$
• C. $a+b+c$
• D. $abc$

Answer 1

Answer: (C)

$a+b+c$

$f\left(x\right)=0\Rightarrow a{x}^2+bx+c=0$
$\alpha ,\beta$ are the roots of $f\left(x\right)=0$
$\frac{1-\alpha }{\alpha },\frac{1-\beta }{\beta }$ are the roots of $p{x}^2+qx+c=0$
Let $\frac{1-\alpha }{\alpha }=x$
$\Rightarrow 1-\alpha =\alpha x$
$\Rightarrow \alpha x+\alpha =1$
$\Rightarrow \alpha (1+x)=1\Rightarrow \alpha =\frac{1}{1+x}$
$f\left(\frac{1}{1+x}\right)=0\Rightarrow a{\left(\frac{1}{1+x}\right)}^2+\frac{b}{1+x}+c=0$
$\Rightarrow a+b(1+x)+c{(1+x)}^2=0$
$r=a+b+c$