若复数x+yi=(1+cosθ)+(t-cos2θ)i(其中x,y,θ∈R),且点(x,y)在抛物线y=x2上,试

若复数x+yi=(1+cosθ)+(t-cos2θ)i(其中x,y,θ∈R),且点(x,y)在抛物线y=x2上,试求实数t的最大值与最小值.

答案

分析:本题以复数的几何意义为载体考查如何建立目标函数及求函数的最值问题.

解:根据两个复数相等的条件,得       

因为点(x,y)在抛物线上,所以t-cos2θ=(1+cosθ)2.    

t=(1+cosθ)2+cos2θ=1+2cosθ+cos2θ+2cos2θ-1=3cos2θ+2cosθ=3(cosθ+)2-.

由于cosθ∈[-1,1],

所以当cosθ=-时,t有最小值-;

当cosθ=1时,t有最大值5.

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