实数x、y满足4x^2-5xy+4y^2=5,设S=x^2+y^2,求S的最值

实数x、y满足4x^2-5xy+4y^2=5,设S=x^2+y^2,求S的最值
实数x、y满足4x^2-5xy+4y^2=5,设S=x^2+y^2,求S的最值

实数x、y满足4x^2-5xy+4y^2=5,设S=x^2+y^2,求S的最值
x=√s cosB y=√s sinB
4x^2-5xy+4y^2=5
4(√s cosB)^2-5√s cosB*√s sinB+4(√s sinB)^2=5
4s (cosB)^2-5s sinBcosB+4s (sinB)^2=5
4s-5s/2 sin2B=5
因为：-1《sin2B《1
所以：s=5/(4-5/2 sin2B)∈[10/13,10/3]
所以s的最大值为：10/3,最小值为：10/13

-s/2<=xy<=s/2
-5s/2<=-5xy<=5s/2
4s-5s/2<=5,4s+5s/2>=5
3s/2<=5,13s/2>=5
10/13<=s<=10/3,