若X、Y、Z满足2/(x-1)=3/(y+1)=4/(z-2),求X、Y、Z为何值时,x^2+y^2-z^2有最大值?并求出这个最大值.

题目

若X、Y、Z满足2/(x-1)=3/(y+1)=4/(z-2),求X、Y、Z为何值时,x^2+y^2-z^2有最大值?并求出这个最大值.

答案

设2/(x-1)=3/(y+1)=4/(z-2)=1/k
则x^2+y^2-z^2
=(2k+1)^2+(3k-1)^2-(4k+2)^2
=-3k^2-18k-2
=-3(k+9/2)^2+235/4
则当k=-9/2时,x^2+y^2-z^2取得最大值为235/4.