Processing math: 4%

Pages

Bookmark and Share

Friday, September 19, 2014

Inverse distances to points

Let P and Q be two distinct points on a plane. For any given point X \neq P,Q, define two functions f(X) and g(X) as follows: f(X) = \frac{1}{PX} + \frac{2}{QX} g(X) = \frac{2}{PX} + \frac{1}{QX} Now for any positive number t, let S(t) be the set of all points Y such that f(Y) \leq tg(Y).

Prove that S(t) is bounded

Prove that S(t) is convex. That is, if A,B are in S(t) then AB is in S(t).

Identify all points X on S(t) such that PX+QX is minimum.

Prove that the area of S(t) is a convex function of t.

No comments:

Post a Comment