Given a non degenerate conic $C$ and a point $O\in C$. I generate an angle $\theta$ uniformly in $[-\pi,\pi)$. Now I draw a line with slope $\tan \theta$ through $O$. Given that this line is not the tangent to $C$ at $O$, is it true that the line will intersect the curve at one more point (in addition to $O$ ie) with probability $1$ ?
Probability was needed to exclude cases like $C = \{(x,y)|y=x^2\}, O = (0,0), \theta = \pi/2$