# If α, β are roots of the equation x

Question:

If $\alpha, \beta$ are roots of the equation $x^{2}+l x+m=0$, write an equation whose roots are $-\frac{1}{\alpha}$ and $-\frac{1}{\beta}$.

Solution:

Given equation: $x^{2}+l x+m=0$

Also, $\alpha$ and $\beta$ are the roots of the equation.

Sum of the roots $=\alpha+\beta=\frac{-l}{1}=-l$

Product of the roots $=\alpha \beta=\frac{m}{1}=m$

Now, sum of the roots $=-\frac{1}{\alpha}-\frac{1}{\beta}=-\frac{\alpha+\beta}{\alpha \beta}=-\frac{-l}{m}=\frac{l}{m}$

Product of the roots $=\frac{1}{\alpha \beta}=\frac{1}{m}$

$\therefore x^{2}-(S$ um of the roots $) x+P$ roduct of $t h e$ roots $=0$

$\Rightarrow x^{2}-\frac{l}{m} x+\frac{1}{m}=0$

$\Rightarrow m x^{2}-l x+1=0$

Hence, this is the equation whose roots are $-\frac{1}{\alpha}$ and $-\frac{1}{\beta}$.