A plane is flying in a straight line parallel to the ground at a constant speed of $400 \text{ m}$ per second at an altitude of $4000\text{ m},$ directly above an observer on the ground. After $3$ seconds, as shown in the above diagram, the observer now looks at the plane at an angle of $\theta$ from vertical. If the rate of change of $\theta$ with respect to time measured in seconds at this instant is $\displaystyle{\frac{d\theta}{dt}=\frac{a}{b}},$ where $a$ and $b$ are positive, coprime integers, what is $a+b?$