The ball describes a path $t \mapsto (400+t V_x, 700+t V_y)$, where you can interpret $t$ as time, if you wish. You are trying to find 'times' such that the distance from the path at that time to the center of the circle matches the radius (the possibilities are never, once or twice).
The distance to the center squared (including the square root complicates things) is
$$d^2 = (t V_x-400)^2 + (t V_y +400)^2$$
This is a quadratic in $t$. Solve the equation $d^2=350^2$ for $t$, if it has real solutions it intersects and the solutions are the 'times' it intersects.
If it intersects, you can use the equation for the path to determine where the intersection occurs.