If $α, β$ are roots of the equation $x^{2} + l x + m = 0,$ write an equation whose roots are $- \frac{1}{α} a n d - \frac{1}{β} .$

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Solution

$α a n d β a r e t h e r o o t s$ $x^{2} + l x + m = 0,$ ...(i)

$∴ α + β = \frac{- b}{a} = 1 a n d α β = \frac{c}{a} = m N o w, The quadratic equation whose roots are - \frac{1}{α} a n d \frac{1}{β} i s x^{2} - (s u m o f r o o t s) x + (p r o d u c t o f r o o t s) = 0 \Rightarrow x^{2} - (- \frac{1}{α} + \frac{- 1}{β}) + (- \frac{1}{α,} \frac{- 1}{β}) = 0 \Rightarrow x^{2} + (\frac{α + β}{α β}) x + \frac{1}{α β} = 0 \Rightarrow x^{2} + (\frac{- 1}{m}) x + \frac{1}{m} = 0 \Rightarrow m x^{2} - l x + 1 = 0$

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