The least value of $|a|$ for which $\tan\theta$ and $\cot\theta$ are roots of the equation $x^2+ax+1=0$, is
- $2$
- $1$
- $\dfrac{1} 2$
- $0$
For the quadratic equation x^2 + ax + 1 = 0 to have real roots tan(theta) and cot(theta), its discriminant must be greater than or equal to zero. This gives the inequality a^2 - 4 >= 0, which simplifies to |a| >= 2. Therefore, the minimum value of |a| is 2.
If tan(theta) and cot(theta) are the roots of the quadratic equation x^2 + ax + 1 = 0, their product according to Vieta's formulas is the constant term divided by the leading coefficient, giving tan(theta) * cot(theta) = 1. Because we know tan(theta) * cot(theta) equals 1, the product of the roots is firmly established as 1. The sum of these roots must equal -a, so we can write tan(theta) + cot(theta) = -a. Expressing the left side in terms of sine and cosine gives (sin^2(theta) + cos^2(theta)) / (sin(theta)cos(theta)), which simplifies to 1 / (sin(theta)cos(theta)). Using the double angle identity, this becomes 2 / sin(2(theta)), and since the absolute maximum value of the sine function is 1, the absolute minimum value of 2 / sin(2(theta)) is 2. Therefore, the least value of |a| is 2.