x^+y^=1,求y-2/x-1的取值范围

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x^+y^=1,求y-2/x-1的取值范围
x^+y^=1,求y-2/x-1的取值范围

x^+y^=1,求y-2/x-1的取值范围
令(y-2)/(x-1)=t
y=2+t(x-1),代入x^2+y^2=1：
x^2+{2+t(x-1)}^2 = 1
x^2 +4+4t(x-1) + t^2(x-1)^2 = 1
(t^2+1)x^2 - 2t(t-2)x + (t^2-4t+3) = 0
判别式 ≥ 0
4t^2(t-2)^2 - 4(t^2+1)(t^2-4t+3) ≥ 0
t^2(t-2)^2 - (t^2+1)(t^2-4t+3) ≥ 0
t^4 - 4t^3 + 4t^2 - (t^4-4t^3+4t^2-4t+3) ≥ 0
4t-3 ≥ 0
t ≥ 3/4
即：(y-2)/(x-1) ≥ 3/4

x^2+y^2=1的图像是一个单位圆，y-2/x-1表示从点（1,2）引N条直线与圆产生交集的直线的斜率，其中两条与圆相切的直线的斜率就是取值区域的两端。

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