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一道求取值范圍的題(1)

2023-02-18 09:56 作者:Euarnecom  | 我要投稿

已知實(shí)數(shù)x,y 滿(mǎn)足2x%5E2%2B4xy%2B5y%5E2%3D1,那么x%5E2%2By%5E2的取值范圍是______。



方法 1:令x%5E2%2By%5E2%3Dr%5E2,那么x%3Dr%5Csin%5Ctheta,y%3Dr%5Ccos%5Ctheta,其中%5Ctheta%5Cin%5Cmathbf%7BR%7D。

%5Cbegin%7Balign*%7D%0A2x%5E2%2B4xy%2B5y%5E2%26%3D2r%5E2%5Csin%5E2%5Ctheta%2B4r%5E2%5Csin%5Ctheta%5Ccos%5Ctheta%2B5r%5E2%5Ccos%5E2%5Ctheta%5C%5C%0A%26%3D2r%5E2%2B4r%5E2%5Csin%5Ctheta%5Ccos%5Ctheta%2B3r%5E2%5Ccos%5E2%5Ctheta%5C%5C%0A%26%3D2r%5E2%2B2r%5E2%5Csin2%5Ctheta%2B%5Cfrac%7B3%7D%7B2%7Dr%5E2%5Ccos2%5Ctheta%2B%5Cfrac%7B3%7D%7B2%7Dr%5E2%5C%5C%0A%26%3D2r%5E2%5Csin2%5Ctheta%2B%5Cfrac%7B3%7D%7B2%7Dr%5E2%5Ccos2%5Ctheta%2B%5Cfrac%7B7%7D%7B2%7Dr%5E2%5C%5C%0A%26%3D%5Cfrac%7B5%7D%7B2%7Dr%5E2%5Csin(2%5Ctheta%2B%5Cvarphi)%2B%5Cfrac%7B7%7D%7B2%7Dr%5E2%0A%5Cend%7Balign*%7D

所以有,%5Cfrac%7B5%7D%7B2%7Dr%5E2%5Csin(2%5Ctheta%2B%5Cvarphi)%2B%5Cfrac%7B7%7D%7B2%7Dr%5E2%3D1,其中%5Cvarphi%3D%5Carcsin%5Cfrac%7B3%7D%7B5%7D%3D%5Carccos%5Cfrac%7B4%7D%7B5%7D。于是,r%5E2%3D%5Cdfrac%7B2%7D%7B5%5Csin(2%5Ctheta%2B%5Cvarphi)%2B7%7D,由于2%5Ctheta%2B%5Cvarphi取值是任意實(shí)數(shù),所以

x%5E2%2By%5E2的取值范圍是%5B%5Cfrac%7B1%7D%7B6%7D%2C1%5D


方法 2:由于2x%5E2%2B4xy%2B5y%5E2%3D1,所以

x%5E2%2By%5E2%3D%5Cfrac%7Bx%5E2%2By%5E2%7D%7B2x%5E2%2B4xy%2B5y%5E2%7D%3D%5Cfrac%7B%5Cdfrac%7Bx%7D%7By%7D%2B%5Cdfrac%7By%7D%7Bx%7D%7D%7B%5Cdfrac%7B2x%7D%7By%7D%2B%5Cdfrac%7B5y%7D%7Bx%7D%2B4%7D

設(shè)t%3D%5Cfrac%7Bx%7D%7By%7D,顯然t%5Cne%200,那么,

x%5E2%2By%5E2%3D%5Cfrac%7B%5Cdfrac%7Bx%7D%7By%7D%2B%5Cdfrac%7By%7D%7Bx%7D%7D%7B%5Cdfrac%7B2x%7D%7By%7D%2B%5Cdfrac%7B5y%7D%7Bx%7D%2B4%7D%3D%5Cfrac%7Bt%2B%5Cdfrac%7B1%7D%7Bt%7D%7D%7B2t%2B%5Cdfrac%7B5%7D%7Bt%7D%2B4%7D%3D%5Cfrac%7Bt%5E2%2B1%7D%7B2t%5E2%2B4t%2B5%7D

x%5E2%2By%5E2的取值范圍是%5B%5Cfrac%7B1%7D%7B6%7D%2C1%5D。

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