HMMT 二月 2008 · 冲刺赛 · 第 29 题
HMMT February 2008 — Guts Round — Problem 29
题目详情
英文原题
- [ 15 ] Let ( x, y ) be a pair of real numbers satisfying − y x
56 x + 33 y = , and 33 x − 56 y = .
2 2 2 2
x + y x + y
Determine the value of | x | + | y | .
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解析
英文解析
- [ 15 ] Let ( x, y ) be a pair of real numbers satisfying − y x
56 x + 33 y = , and 33 x − 56 y = .
2 2 2 2
x + y x + y
Determine the value of | x | + | y | .
Answer: Observe that 11
1 x − yi 65 = = 33 x − 56 y + (56 x + 33 y ) i = (33 + 56 i )( x + yi ) .
2 2
x + yi x + y
( )So
1 1 7 − 4 i 2
( x + yi ) = = = .2
33 + 56 i (7 + 4 i ) 652
( )
7 4
It follows that ( x, y ) = ± , − .
65 65
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