Time when two particles will meet Particle P moves with constant speed of 13m/s always aiming at O . As P starts moving , O moves with constant velocity of 5m/s towards positive y-axis. Time taken by particle P to catch O will be?

I tried solving this by realtive motion concept but i am not able to find the path it will follow.
Please help me out with this problem....
 A: The solution to this type of problem is called a "pursuit curve" and is generally found by constructing and solving a differential equation for the co-ordinates of the pursuer as a function of time. In this particular case the pursuit curve is called a radiodrome.
If the co-ordinates of $O$ at time $t$ are $(x_O(t), y_O(t))$ (we assume that the functions $x_O(t)$ and $y_O(t)$ are known) and the co-ordinates of $P$ at time $t$ are $(x_P(t), y_P(t))$ then the condition that $P$ always aims at $O$ is
$\frac{dy_P}{dx_P} = \frac{y_O-y_P}{x_O-x_P}$
and the condition that $P$ travels at constant speed $s_P$ is
$\left(\frac{dy_P}{dt}\right)^2 + \left(\frac{dx_P}{dt}\right)^2 = (s_P)^2$
A useful text about pursuit curves in general is Paul Nahin's Chases and Escapes.
Note that the calculation of the pursuit curve is more complex that the related problem where $P$ anticipates that $O$ is travelling at a constant velocity and calculates an interception course, which will be a straight line. On the interception course $P$ is not aiming at $O$ itself but at the point where $O$ will be at a future time.
