# Finding Point of Intersection with Circle

I have a circle with center at point $(800,300)$ and radius of $350$. I have a ball at $(400,700)$ and traveling at velocity $(v_x, v_y)$.

How can I find where the ball will intercept the circle?

I'm doing this in code, but just really trying to figure out the math.

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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.
And as the question was "where", not "when", you (OP) will still have one step to do once you have found $t$. –  Gerry Myerson Apr 25 '12 at 4:03