HMMT 二月 2007 · 代数 · 第 5 题
HMMT February 2007 — Algebra — Problem 5
题目详情
- [ 5 ] A convex quadrilateral is determined by the points of intersection of the curves x + y = 100 and xy = 4; determine its area. 3 2
解析
- [ 5 ] A convex quadrilateral is determined by the points of intersection of the curves x + y = 100 and xy = 4; determine its area. √ Answer: 4 17 . By symmetry, the quadrilateral is a rectangle having x = y and x = − y as axes of symmetry. Let ( a, b ) with a > b > 0 be one of the vertices. Then the desired area is ( ) ( ) √ √ √ √ √ 2 2 4 2 2 4 2 2( a − b ) · 2( a + b ) = 2( a − b ) = 2 a − 2 a b + b = 2 100 − 2 · 4 = 4 17 . 3 2