作业帮若实数x,y满足2x^2+y^2=4x,求S=x^2+y^2的值域.[0,4]
来源：学生作业帮助网 编辑：作业帮 时间：2024/11/24 02:33:54
![若实数x,y满足2x^2+y^2=4x,求S=x^2+y^2的值域.[0,4]](/uploads/image/z/8730606-30-6.jpg?t=%E8%8B%A5%E5%AE%9E%E6%95%B0x%2Cy%E6%BB%A1%E8%B6%B32x%5E2%2By%5E2%3D4x%2C%E6%B1%82S%3Dx%5E2%2By%5E2%E7%9A%84%E5%80%BC%E5%9F%9F.%5B0%2C4%5D)
xOO0ƿJLֱ } F⟨ф $#E! Ɩy+u-lJkqɇysjY;⽣ܯ2o=" ʀK=bKf(DΕbS`);K 1ŪrdcPnʪ@Th^M]3 dj^+P9*cY֦1Yˆu65Ĉ+}`8Fyؘ$ LV|Ϻ:/Yg\ ~W}8|l|קS>lb!*SDc~]J
若实数x,y满足2x^2+y^2=4x,求S=x^2+y^2的值域.[0,4]
若实数x,y满足2x^2+y^2=4x,求S=x^2+y^2的值域.
[0,4]
若实数x,y满足2x^2+y^2=4x,求S=x^2+y^2的值域.[0,4]
y^2=-2x^2+4x>=0
x(x-2)
2x^2+y^2=4x
2(x-1)^2+y^2=2
可设x = cosa + 1, y = 根号2 * sina
S = (cosa + 1)^2 + 2 * (sina)^2
= (cosa)^2 + 2 * cosa + 1 + 2 * (1 - (cosa)^2)
= -(cosa)^2 + 2 * cosa + 3
= -(cosa - 1...
全部展开
2x^2+y^2=4x
2(x-1)^2+y^2=2
可设x = cosa + 1, y = 根号2 * sina
S = (cosa + 1)^2 + 2 * (sina)^2
= (cosa)^2 + 2 * cosa + 1 + 2 * (1 - (cosa)^2)
= -(cosa)^2 + 2 * cosa + 3
= -(cosa - 1)^2 + 4
-1<=cosa<=1
-2<=cosa-1<=0
0<=(cosa - 1)^2<=4
-4<=-(cosa - 1)^2<=0
0<=-(cosa - 1)^2+4<=4
所以答案是[0,4]
收起
y^2=-2x^2+4x>=0
x(x-2)<=0
0<=x<=2
x^2+y^2=x^2-2x^2+4x=-x^2+4x=-(x-2)^2+4
0<=x<=2，在x=2左边，是增函数
所以x=0,最小=0
x=2，最大=4
所以[0,4]