HMMT 二月 2000 · 团队赛 · 第 12 题

HMMT February 2000 — Team Round — Problem 12

A t a dan e, Abhina v starts from p oin t ( a; 0) and mo v es along the negativ e x dire tion with sp eed v , while P ei-Hsin starts from (0 ; b ) and glides in the negativ e y-dire tion a with sp eed v . What is the distan e of losest approa h b et w een the t w o? b

解析

A 's p osition is ( a V t; 0) and P 's p osition is (0 ; b V t ). So, at time t , the distan e a b q 2 2 b et w een them is ( a V t ) + ( b V t ) . Noti e that this distan e is the same as the a b distan e b et w een the p oin t ( a; b ) and the line ( V t; V t ), whi h is the same as the line a b j Ax + B y + C j 0 0 p V x V y = 0. The distan e from a line Ax + B y + C = 0 and ( x ; y ) is , b a 0 0 2 2 A + B j bV aV j a b p so the answ er is 2 2 V + V a b