Question

If $a\in [-5,30]$, then the probability that the graph of the function $y=x^2+2(a+4)x-5a+64$ is strictly above the x-axis is

• A. $\frac{27}{35}$
• B. $\frac{8}{25}$
• C. $\frac{8}{35}$
• D. $\frac{17}{25}$

Answer 1

Answer: (C)

$\frac{8}{35}$

The total length of the interval in which $a$ lies $=30-(-5)=35$
If the graph of $y=x^2+2(a+4)x-5a+64$ is entirely above the x-axis, the discriminant of the above quadratic expression must be negative.
$\therefore 4{(a+4)}^2+4(5a-64)<0\Rightarrow a^2+13a-48<0\Rightarrow (a+16)(a-3)<0\Rightarrow -16<a<3$
But $a\in [-5,30]\therefore -5<a<3$ for the evnt to happen.
the length of this interval $=3-(-5)=8$
Hence, the required probability $\frac{8}{35}$