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Common Roots

lf α, β are...

Question

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

A

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

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B

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

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C

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

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D

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

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Solution

The correct option is A $x^{2} + 6 x + 9 = 0$
Equation with roots having reciprocals of given
equation is $f (\frac{1}{α}) = 0$
$\Rightarrow 9 (\frac{1}{α})^{2} + 6 (\frac{1}{α}) + 1 = 0$
$\Rightarrow 9 + 6 α + α^{2} = 0$
$∴ x^{2} + 6 x + 9 = 0$ is the required equation

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