若x,y满足3x^2+2y^2=6x,求z=x^2+y^2值域

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若x,y满足3x^2+2y^2=6x,求z=x^2+y^2值域
若x,y满足3x^2+2y^2=6x,求z=x^2+y^2值域

若x,y满足3x^2+2y^2=6x,求z=x^2+y^2值域
3x^2+2y^2=6x
6x>=0
x>=0
3x^2+2y^2=6x
2x^2+2y^2=-x^2+6x
x^2+y^2=-x^2+6x/2
z=x^2+y^2
z=-x^2+6x/2=-1/2(x^2-6x+9-9)
=-1/2(x-3)^2+9/2

3x^2+2y^2=6x
3(x-1)²+2y²=3
(x-1)²+y²/(3/2)=1
可设x=cost+1 y=(√6/2)sint
于是z=(cost+1)²+(√6/2*sint)²
=(-1/2)(cost-2)²+9/2
所以z最大=-1/2+9/2=4 z最小=-9/2+9/2=0
故值域为[0, 4]

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