The roots of a quadratic equation $x^{2} - 4 x - {log}_{3} a = 0$ are real. Then what is the least value of $a$ ?

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$\frac{1}{81}$

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$\frac{1}{64}$

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Solution

The correct option is B
$\frac{1}{81}$

We have, $x^{2} - 4 x - {log}_{3} a = 0$
If roots are real, then $D \geq 0$
$∴ 16 + 4 {log}_{3} a \geq 0$
$\Rightarrow {log}_{3} a \geq - \frac{16}{4} \Rightarrow {log}_{3} a \geq - 4$
$\Rightarrow a \geq 3^{- 4} \Rightarrow a \geq \frac{1}{81}$

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