If $α, β$ are the roots of the equation $l x^{2} + m x + n = 0$ , find the equation whose roots are $\frac{α}{β}, \frac{β}{α}$

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Solution

Given that $α, β$ are roots of equation $l x^{2} + m x + n = 0$
So we have $α + β = - \frac{m}{l}$ and $α β = \frac{n}{l}$
Let us consider $x^{2} + a x + b = 0$ be our required equation whose roots are $\frac{α}{β}, \frac{β}{α}$
Sum of roots is $- a = \frac{α}{β} + \frac{β}{α} = \frac{α^{2} + β^{2}}{α β} = \frac{(α + β)^{2} - 2 α β}{α β} = \frac{m^{2} - 2 n l}{n l}$
$\Rightarrow a = - (\frac{m^{2} - 2 n l}{n l})$
Product of roots is $b = \frac{α}{β} \times \frac{β}{α} = 1$
Therefore the required equation is $x^{2} - (\frac{m^{2} - 2 n l}{n l}) x + 1 = 0$
$\Rightarrow n l x^{2} - (m^{2} - 2 n l) x + n l = 0$

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