推 tallguyshen :題目不是規定a,b,c為相異實數嗎? 1會得到a=b=c 07/11 22:19
→ tallguyshen :2 會得到a=b=c=0 這樣不是矛盾嗎?? 07/11 22:20
※ 引述《ccccc7784 (龍王號)》之銘言：
: 請問這題應如何做??
: http://i.imgur.com/6XbXYvi.png
: 答案是給3或-3/8
: 謝謝
a + b b + c c + a
abc≠0且a , b , c 為相異實數 , 若 ------- = ------- = ------- ,
c a b
3abc
則 ----------------------- = ？
(a+b-c)(b+c-a)(c+a-b)
解: 1. a + b = b + c = c + a ≠ 0
a + b b + c c + a
令 ------- = ------- = ------- = r ≠0
c a b
則 a + b = cr ------(1)
b + c = ar ------(2)
c + a = br ------(3)
(1) + (2) + (3) => (2)(a + b + c) = (a + b + c)(r) => r = 2
a + b = 2c => a + b - c = c
b + c = 2a => b + c - a = a
c + a = 2b => c + a - b = b
3abc
-----------------------------------
(a + b - c)(b + c - a)(c + a - b)
3abc
= ----------- = 3
(c)(a)(b)
2 . a + b = b + c = c + a = 0
a + b = 0
b + c = 0
c + a = 0
三式相加 (2)(a + b + c) = 0 => a + b + c = 0
∴ a + b = -c , b + c = -a , c + a = -b
3abc
----------------------------------
(a + b - c)(b + c - a)(c + a - b)
3abc
= --------------------------
(-c - c)(-a - a)(-b - b)
3abc -3
= ------------------ = ---
(-2c)(-2a)(-2b) 8
由 1 和 2 得
3abc -3
----------------------------------- = 3 或 ---
(a + b - c)(b + c - a)(c + a - b) 8
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※ 編輯: LuisSantos (114.36.163.166), 07/11/2014 19:51:55