已知 a＞b＞0,a²＋b²＝4ab,求a－b／a+b的值

Question

已知 a＞b＞0,a²＋b²＝4ab,求a－b／a+b的值

Answer 1

法一:
a－b／a+b
=(a+b-2b)/(a+b)
=1-2/(a/b+1)
∵a²＋b²＝4ab
∴(a/b)²+1=4a/b
∴(a/b)²-4a/b+1=0
△=4²-4=12
a/b=(4±2√3)/2=2±√3
∴原式=1-2/(2+√3+1)
=1-2/(3+√3)
=1-2(3-√3)/(9-3)
=1-(3-√3)/3
=√3/3
或原式=1-2/(2-√3+1)
=1-2/(3-√3)
=1-2(3+√3)/(9-3)
=1-(3+√3)/3
=-√3/3
∴原式=±√3/3
法二:(a-b)/(a+b)
=±√(a-b)²/(a+b)²
=±√(a²+b²-2ab)/(a²+b²+2ab)
=±√(2ab/6ab)
=±√3/3
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