Question

If $\alpha$ and $\beta$ are the roots of the equation $a{x}^2+bx+c=0$. Find the equation whose roots are as given below.
$\frac{1}{\alpha +\beta },\frac{1}{\alpha }+\frac{1}{\beta }$

Answer 1

$\alpha +\beta =-\frac{b}{a},\alpha \beta =\frac{c}{a}$
$sum=\frac{1}{\alpha +\beta }.\frac{\alpha +\beta }{\alpha \beta }=\frac{a}{b}-\frac{a}{c}=\frac{(ac+b^2)}{bc}$
Product $=\frac{1}{\alpha +\beta }.\frac{\alpha +\beta }{\alpha \beta }=\frac{1}{\alpha \beta }=\frac{a}{c}$
$\therefore$ Equation is $x^2-Sx+p=0$
or $x^2+\frac{ac+b^2}{bc}x+\frac{a}{c}=0$
or $bc{x}^2+(b^2+ac)x+ab=0$