CMR: $a^{2}+b^{2}=1$
Edited by misakichan, 03-09-2016 - 14:02.
Cho a, b thỏa mãn: $a\sqrt{1-b^{2}}+b\sqrt{1-a^{2}}=1$ CMR: $a^{2}+b^{2}=1$
Started By misakichan, 03-09-2016 - 13:59
#1
Posted 03-09-2016 - 13:59
misakichan
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Trung sĩ
Cho a, b thỏa mãn: $a\sqrt{1-b^{2}}+b\sqrt{1-a^{2}}=1$
CMR: $a^{2}+b^{2}=1$
CMR: $a^{2}+b^{2}=1$
#2
Posted 03-09-2016 - 14:20
Baoriven
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Thượng úy
Ta chỉ cần dùng AM-GM.
Ta có: $a\sqrt{1-b^{2}}+b\sqrt{1-a^{2}}\leq \frac{a^2+1-b^2}{2}+\frac{b^2+1-a^2}{2}=1$
Dấu bằng xảy ra khi: $a^2+b^2=1$.
Edited by Baoriven, 03-09-2016 - 14:24.
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