Question

已知a∈R,問復(fù)數(shù)z=(a²-2a+4)-(a²-2a+2)i所對(duì)應(yīng)的點(diǎn)在第幾象限？復(fù)數(shù)z對(duì)應(yīng)的點(diǎn)的軌跡是什么？

Answer 1

思路分析:根據(jù)復(fù)數(shù)與復(fù)平面上點(diǎn)的對(duì)應(yīng)關(guān)系,知復(fù)數(shù)z對(duì)應(yīng)的點(diǎn)在第幾象限與復(fù)數(shù)z的實(shí)部和虛部有關(guān).所以本題的關(guān)鍵是判斷(a²-2a+4)與-(a²-2a+2)的符號(hào).求復(fù)數(shù)z對(duì)應(yīng)點(diǎn)的軌跡問題,首先把z表示成z=x+yi(x,y∈R)的形式,然后尋求x、y之間的關(guān)系,但要注意參數(shù)限定的條件.

解:由a²-2a+4=(a-1)²+3≥3,-(a²-2a+2)=-(a-1)²-1≤-1,

得z的實(shí)部為正數(shù),z的虛部為負(fù)數(shù).

∴復(fù)數(shù)z的對(duì)應(yīng)點(diǎn)在第四象限.

設(shè)z=x+yi(x,y∈R),則

消去a²-2a,得y=-x+2(x≥3).∴復(fù)數(shù)z的對(duì)應(yīng)點(diǎn)的軌跡是一條直線,其方程為y=-x+2(x≥3).