Question

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

Answer 1

Given equation:

\[x^2 + lx + m = 0\]

Also, 

\[\alpha \text { and } \beta\] 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 = \[\frac{1}{\alpha\beta} = \frac{1}{m}\]

\[\therefore x^2 - \left( \text { Sum of the roots } \right)x +\text {  Product of the roots } = 0\]

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

\[ \Rightarrow m x^2 - lx + 1 = 0\]

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