Given array of line segments (line between two points). Find can a line (infinite length) be drawn from origin(0,0) which does not intersect any of these lines segements.

line: p1(x1,y1) to p2(x2,y2)
Return type: True or False

Follow up:

Find if a line can be drawn passing through point (x,y) without intersecting any other line segments.

I solved it in O(n*logn) time complexcity with some help from interviewer.
Problem boils down to finding gaps in a list of pairs.

Algo:

1. Find angle formed by line with origin. [Interviewer mentioned that I can assume few mathematical functions if needed]. (Returns value of angle in radian)
2. Convert array of line segments to array of angles(list of startAngle, endAngle).
3. For case when line passes through one gap but doesn't pass from other side, he suggested we can add complement of each angle into array as well. [Complement of sector(angle1, angle2) will be sector((angle1+pi)%pi, (angle2+pe)%pi)]
4. Check can we find a gap between two segment in the angles array.

Follow up soln:
Move origin to point given. (Shifiting origin)