Question

If a $\in [-6,12]$, the probability that graph of $y=x^2+2(a+4)x+(3a+40)$ is strictly below x-axis is

• A. $\frac{2}{3}$
• B. $\frac{1}{3}$
• C. $\frac{1}{2}$
• D. $\frac{1}{4}$

Answer 1

The correct option is C $\frac{1}{2}$

The graph is below x-axis means that the roots are imaginary, i.e. the discriminant is negative.
Thus, ${\left\{2\right(a+4\left)\right\}}^2-4\left(1\right)(3a+40)<0$
Or, $-8<a<3$
But, $a\in [-6,12]$
Hence, the overlap is $a\in [-6,3]$
Hence, the required probability = $\frac{lengthoffavourableline}{totallengthofline}=\frac{9}{18}=\frac{1}{2}$