First slide

Theory of equations

Question

If $α, β$ are the roots of $9 x^{2} + 6 x + 1 = 0$ then the equation with the roots $\frac{1}{α}, \frac{1}{β}$ is

Moderate

A

$2 x^{2} + 3 x + 18 = 0$
B

$x^{2} + 6 x - 9 = 0$
C

$x^{2} + 6 x + 9 = 0$
D

$x^{2} - 6 x + 9 = 0$

Solution

Given equation is $9 x^{2} + 6 x + 1 = 0$
$\Rightarrow α + β = \frac{- 6}{9} = \frac{- 2}{3}$ and $αβ = 1 / 9$
$∴ α - β = \sqrt{{(α + β)}^{2} - 4 αβ} = \sqrt{\frac{4}{9} - 4 . \frac{1}{9}} = 0$
$\Rightarrow α = \frac{- 1}{3}, β = \frac{- 1}{3}$
$∴ Equation x^{2} - (α + β) x + αβ = 0$
$\Rightarrow x^{2} + 6 x + 9 = 0$
(or)
$f (\frac{1}{x}) = 0 \Rightarrow x^{2} + 6 x + 9 = 0$

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