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  4. If tan θ and cot θ are roots of x2+2 a x+b=0, then least value of |a| is

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Q. If $\tan \theta$ and $\cot \theta$ are roots of $x^2+2 a x+b=0$, then least value of $|a|$ is

Complex Numbers and Quadratic Equations

Solution:

$\tan \theta+\cot \theta=2 a$ and $\tan \theta \cot \theta=b$
Now, $2 a=\tan \theta+\cot \theta \geq 2$ if $\tan \theta>0$
and $2 a=\tan \theta+\cot \theta \leq-2$ if $\tan \theta<0$.
$\Rightarrow 2|a| \geq 2 \Rightarrow|a| \geq 1$
Thus, least value of $|a|$ is 1 .