### Geometry Problem : Line of Sight

**Source:**IBM Ponder This November 2012

**Problem:**

As the trees grow, more and more points outside the circle of trees stop having a direct line of sight with the origin. What will be the trees' radius when the origin first loses its line of sight with all the points outside the circle? Please give your answer as a decimal number with an accuracy of 13 digits (13 significant digits).

The image is a sketch of a forest of radius 5 and a light beam entering the origin (center of the forest).

Line of sight between the point A = (70,69) and B = (70,70) passing through (0,0)and equidistant from both A and B. Will be the last standing line of sight.

ReplyDeleteNo the above solution is wrong

ReplyDeleteLine of sight between the point A = (1,0) and B = (98,1) passing through (0,0)and equidistant from both A and B. Will be the last standing line of sight.

I did a miscalculation

can you please explain how you arrived at those two points

ReplyDeleteInteersting post but i am not getting the answer for this puzzle.. Abacus may solve it but how it may implement..Please help me to know.

ReplyDeleteAnswer as per IBM Ponder This:

ReplyDeleteThe answer is 1/sqrt(d), where d is the smallest integer satisfying d=a^2+b^2, where a is co-prime to b, and d>=R^2.

Clearly, d<=R^2+1 by choosing a=R and b=1; and d can be R if and only if all the prime factors of R are 1 modulo 4.

In our case, R=9801=3^4*11^2, so the critical radius is 1/sqrt(R^2+1) which is 0.000102030404529629...

Can anyone decipher the solution?