Question

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

• A. None of these
• B. $ac{x}^2+(a+c)cx+(a+c{)}^2=0$
• C. $ab{x}^2+(a+c)bx+(a+c{)}^2=0$
• D. $ac{x}^2+(a+c)bx+(a+c{)}^2=0$

Answer 1

The correct option is D $ac{x}^2+(a+c)bx+(a+c{)}^2=0$

Given: $a{x}^2+bx+c=0$ has roots $\alpha ,\beta$.

Here $\alpha +\beta =-\frac{b}{a}$ and $\alpha \beta =\frac{c}{a}$

If roots are $\alpha +\frac{1}{\beta },\beta +\frac{1}{\alpha },$ then sum of roots are
$=(\alpha +\frac{1}{\beta })+(\beta +\frac{1}{\alpha })=(\alpha +\beta )+\frac{\alpha +\beta }{\alpha \beta }=-\frac{b}{ac}(a+c)$
and product $=(\alpha +\frac{1}{\beta })(\beta +\frac{1}{a})$

$=\alpha \beta +1+1+\frac{1}{\alpha \beta }=2+\frac{c}{a}+\frac{a}{c}=\frac{2ac+c^2+a^2}{ac}=\frac{(a+c{)}^2}{ac}$

Hence required equation is given by
$x^2+\frac{b}{ac}(a+c)x+\frac{(a+c{)}^2}{ac}=0$

$\Rightarrow ac{x}^2+(a+c)bx+(a+c{)}^2=0$