Question

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

Answer 1

Given equation: $x^2+lx+m=0$
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 = $\frac{1}{\alpha \beta }=\frac{1}{m}$

$$\begin{gathered} \therefore x^2-\left(S\mathrm{um}\mathrm{of}the\mathrm{roots}\right)x+P\mathrm{roduct}\mathrm{of}the\mathrm{roots}=0 \\ \Rightarrow x^2-\frac{l}{m}x+\frac{1}{m}=0 \\ \Rightarrow m{x}^2-lx+1=0 \end{gathered}$$

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