# The function of distance between two points with time

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. • its a more complex problem. this is just a part of that. I want to know the increase in distance between p and z(t) with time t. – CODError Jul 21 '16 at 9:03
• That doesn't answer my question. What have you tried? – Jossie Calderon Jul 21 '16 at 9:06
• DO you mean my approach to this? – CODError Jul 21 '16 at 9:07
• Yes @CODError.. – Jossie Calderon Jul 21 '16 at 9:12
• I have put some extra details. I couldnt think of a proper approach. I am stuck with the details I have provided. Now, my main concern is to prove the the dark line curve could not go above dashed line shown in the example graph. I am actually doing a linear interpolation, and I dont want the actual distance value to have higher value that the linearly interpolated value at any time t – CODError Jul 21 '16 at 9:28

Use coordinate geometry. The line L is along the $x$-axis, the point $P(0,p)$ is on the $y$-axis, $Q(a,b)$ is the second point and the radius $r= kt$. The equation of the circle $$(x-a)^2+(y-b)^2=k^2t^2$$ Put $y=0$ and solve for $x$ to get $$z(t)=a+\sqrt{k^2t^2-b^2}$$ and the distance $d$ is given by $$d^2=p^2+\left(a+\sqrt{k^2t^2-b^2}\right)^2$$ Special case: $p=5, a=4, b=3, k=1$
$$d=\sqrt{25+\left(4+\sqrt{t^2-9}\right)^2}$$
This curve is above the dotted line for $3\leq t \leq 10$