Any hint will be appreciated
11322 - Romeo and Juliet
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I was about to think, that area is an ellipse, how did you find out that the area is a circle?sclo wrote:The regions where Romeo and Juliet can meet is actually a circle with a circular hole. (except for a boundary case where one circle is a halfplane)
The entire problem can be reduced to finding intersections of circles.
Set up an equation as follows, without loss of generality, assume both of the them are initially on the x-axis. Romeo is at (0,0) and Juliet is at (h,0).Leonid wrote:I was about to think, that area is an ellipse, how did you find out that the area is a circle?sclo wrote:The regions where Romeo and Juliet can meet is actually a circle with a circular hole. (except for a boundary case where one circle is a halfplane)
The entire problem can be reduced to finding intersections of circles.
Also, suppose Romeo walks at speed u1 and Juliet walks at speed u2.
When they meet at (x,y), they have to meet at same time, so we obtain equation d(x-0,y-0)/u1 = d(x-h,y-0)/u2 where d(a,b)=sqrt(a^2+b^2).
Squaring both side and simplifying, we see that we get Ax^2+Bx+Ay^2+C=0 for some constants A,B,C. Since coefficients of x^2 and y^2 are same, we conclude that it is a circle. Notice that in the degenerate case that u1=u2, we get A=0. That case needs to be handled specially.