已知向量a,b满足|a|=|b|=1,且|ka+b|=根号3|a-kb|(k>0),令f(k)=a*b(1)求f(k)=a*b(用k表示)(2)当k>o时,f(k)≥x^2-2tx-1/2对任意的t∈{-1,1}恒成立,求实数x的取值范围
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![已知向量a,b满足|a|=|b|=1,且|ka+b|=根号3|a-kb|(k>0),令f(k)=a*b(1)求f(k)=a*b(用k表示)(2)当k>o时,f(k)≥x^2-2tx-1/2对任意的t∈{-1,1}恒成立,求实数x的取值范围](/uploads/image/z/4036240-64-0.jpg?t=%E5%B7%B2%E7%9F%A5%E5%90%91%E9%87%8Fa%2Cb%E6%BB%A1%E8%B6%B3%7Ca%7C%3D%7Cb%7C%3D1%2C%E4%B8%94%7Cka%2Bb%7C%3D%E6%A0%B9%E5%8F%B73%7Ca-kb%7C%28k%3E0%29%2C%E4%BB%A4f%28k%29%3Da%2Ab%281%29%E6%B1%82f%28k%29%3Da%2Ab%EF%BC%88%E7%94%A8k%E8%A1%A8%E7%A4%BA%EF%BC%89%EF%BC%882%EF%BC%89%E5%BD%93k%3Eo%E6%97%B6%2Cf%EF%BC%88k%EF%BC%89%E2%89%A5x%5E2-2tx-1%2F2%E5%AF%B9%E4%BB%BB%E6%84%8F%E7%9A%84t%E2%88%88%7B-1%2C1%7D%E6%81%92%E6%88%90%E7%AB%8B%2C%E6%B1%82%E5%AE%9E%E6%95%B0x%E7%9A%84%E5%8F%96%E5%80%BC%E8%8C%83%E5%9B%B4)
已知向量a,b满足|a|=|b|=1,且|ka+b|=根号3|a-kb|(k>0),令f(k)=a*b(1)求f(k)=a*b(用k表示)(2)当k>o时,f(k)≥x^2-2tx-1/2对任意的t∈{-1,1}恒成立,求实数x的取值范围
已知向量a,b满足|a|=|b|=1,且|ka+b|=根号3|a-kb|(k>0),令f(k)=a*b(1)求f(k)=a*b(用k表示)(2)当k>o时,
f(k)≥x^2-2tx-1/2对任意的t∈{-1,1}恒成立,求实数x的取值范围
已知向量a,b满足|a|=|b|=1,且|ka+b|=根号3|a-kb|(k>0),令f(k)=a*b(1)求f(k)=a*b(用k表示)(2)当k>o时,f(k)≥x^2-2tx-1/2对任意的t∈{-1,1}恒成立,求实数x的取值范围
(1)设a=(x1,y1),b=(x2,y2)
ka+b=(kx1+x2,ky1+y2)
a-kb=(x1-kx2,y1-ky2)
F(k)=a*b=(x1x2,y1y2)
|a|=√(x1^2+y1^2)=1=>x1^2+y1^2=1
|b|=√(x2^2+y2^2)=1=>x2^2+y2^2=1
|ka+b|=√[(kx1+x2)^2+(ky1+y2)^2]
|a-kb|=√[(x1-kx2)^2+(y1-ky2)^2]
|ka+b|=根号3*|a-kb|
=>(kx1+x2)^2+(ky1+y2)^2=3[(x1-kx2)^2+(y1-ky2)^2]
=>(k^2-3)(x1^2+y1^2)+(1-3k^2)(x2^2+y2^2)+8k(x1x2+y1y2)=0
=>x1x2+y1y2=(1+k^2)/(4k)
=>F(k)=(1+k^2)/(4k)
(2)F(k)=(1+k^2)/(4k)=(1-k)^2/(4k)+1/2
当k=1,F(k)有最小值1/2.因此
x^2-2tx-1/2≤1/2
x^2-2tx-1≤0
t-√(t^2+1)≤x≤t+√(t^2+1)
∵t∈[-1,1]
∴-1-√2≤x≤1+√2
|ka+b|=√3|a-kb|
∴k∧2*a∧2+2ka*b+b∧2=3(a∧2-2kab+k∧2*b∧2)
∴k∧2+2kab+1=3(1-2kab+k∧2)
∴8kab=2k∧2+2
∴ab=(2k∧2+2)/8k
∴ab=(1 k∧2)/4k
∴f(k)=(1 k∧2)/4k