Consider I have two points p and q, and a line segment l: y=mx+c (actually the enpoints of the segment are given). There is a circle with center q which is growing with time t, i.e. the radius r = k.t where k is some constant. Consider z(t) be a point(s) of intersection between the line segment and the growing circle.
What would be the shape of the graph between d(p, z(t)) and t, where d(p, z(t)) is the distance between point p and z(t) . we take the intersection point z(t) which is far from p.
I can find the intersection points z(t) at any time t because the radius and the center are known. Then I simply calculate the distance between p and z(t). I get the intuition that the graph will be similar to the dark line shown in picture below. Is it possible that the curve can go above the dashed line.