Question

动点P满足向量OP=α向量OM+β向量ON,其中α^2+β^2=1,α,β∈R

Answer 1

【1】略.

点P的轨迹是单位圆,方程为x²+y²=1.

由上面结论,可设点P(cost,sint).(t∈R).

∴向量PM=(-1-cost,-sint),向量PN=(-cost,1-sint).

∴PM•PN=(-1-cost,-sint) •(-cost,1-sint).

=(1+cost)cost+(sint-1)sint

=1+cost-sint

=1-(√2)sin[t-(π/4)].

∴1-√2≤PM•PN≤1+√2.