If $x^{2} + 4 a x + 3 = 0$ and $2 x^{2} + 3 a x - 9 = 0$ have a common root, then $a =$

0

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2

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- 1

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\pm 1

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Solution

The correct option is D $\pm 1$
Let $α$ be the common root.
$\Rightarrow \frac{α^{2}}{b_{1} c_{2} - b_{2} c_{1}} = \frac{α}{c_{1} a_{2} - a_{1} c_{2}} = \frac{1}{a_{1} b_{2} - a_{2} b_{1}}$
$\Rightarrow \frac{α^{2}}{4 a (- 9) - (3 a) 3} = \frac{α}{3 (2) - 1 (- 9)} = \frac{1}{1 (3 a) - 2 (4 a)}$
$\Rightarrow \frac{α^{2}}{- 45 a} = \frac{1}{- 5 a} \Rightarrow α^{2} = 9$
$\Rightarrow \frac{α}{15} = \frac{1}{- 5 a}$

$\Rightarrow α = \frac{- 3}{a}$
$\Rightarrow a = \frac{- 3}{α} = \frac{- 3}{\pm 3} = \pm 1$

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