Question

Find those equations whose roots are (a) reciprocal of the roots of (b) equal in magnitude but opposite in sign to the roots of the equation $a{x}^2+bx+c=0$

Answer 1

Let the roots of $a{x}^2+bx+c=0$

be $\alpha ,\beta$ and

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

(a) reciprocal roots $\Rightarrow \frac{1}{\alpha },\frac{1}{\beta }$

$\therefore$ sum of roots = $\frac{1}{\alpha }+\frac{1}{\beta }=\frac{\alpha +\beta }{\alpha \beta }=\frac{b/a}{c/a}$

$=\frac{-b}{c}$

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

$\therefore$ required equation is

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

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

(b) roots $\Rightarrow -\alpha ,-\beta$

$\therefore$ sum of roots = $-(\alpha +\beta )=-\left(\frac{-b}{a}\right)=\frac{b}{a}$

Product of roots = $(-\alpha )(-\beta )=\alpha \beta =\frac{c}{a}$

$\therefore$ required equation is

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

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