已知O为坐标原点,向量OA=(sinα,1),向量OB=(cosα,9),OC=(-sinα,2),点P满足向量AB=向
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已知O为坐标原点,向量OA=(sinα,1),向量OB=(cosα,9),OC=(-sinα,2),点P满足向量AB=向量BP
1.记函数f(α)=向量PB*向量CA,α∈(-π/8,π/2)讨论函数的单调性并求其值域
2,若OPC三点共线 ,求|向量OA+向量OB|的值
1.记函数f(α)=向量PB*向量CA,α∈(-π/8,π/2)讨论函数的单调性并求其值域
2,若OPC三点共线 ,求|向量OA+向量OB|的值
![已知O为坐标原点,向量OA=(sinα,1),向量OB=(cosα,9),OC=(-sinα,2),点P满足向量AB=向](/uploads/image/z/15834770-26-0.jpg?t=%E5%B7%B2%E7%9F%A5O%E4%B8%BA%E5%9D%90%E6%A0%87%E5%8E%9F%E7%82%B9%2C%E5%90%91%E9%87%8FOA%3D%28sin%CE%B1%2C1%29%2C%E5%90%91%E9%87%8FOB%3D%28cos%CE%B1%2C9%29%2COC%3D%28-sin%CE%B1%2C2%29%2C%E7%82%B9P%E6%BB%A1%E8%B6%B3%E5%90%91%E9%87%8FAB%3D%E5%90%91)
AB
=OB - OA
=( cosα-sinα,8)
f(α)
=PB.CA
= -AB.CA
= -( cosα-sinα,8) . ( -2sinα,1)
=2sinαcosα-2(sinα)^2 + 8
= sin2α +(1-2(sinα)^2) + 7
= sin2α + cos2α + 7
= √2(sin(2α+π/4) + 7
值域 = [7-√2,7+√2]
增加 (-π/8,π/8]
减小[π/8, π/2)
(2)
let P be (x,y)
OPC三点共线
-2/sinα = y/x
-2x = ysinα
x = -ysinα/2
AB= BP
(cosα-sinα, 8) = (-ysinα/2-cosα, y-9)
=> y-9 = 8 and cosα-sinα= -ysinα/2-cosα
= y = 17 and cosα-sinα= -17sinα/2-cosα
cosα-sinα= -17sinα/2-cosα
2cosα = -15sinα/2
cosα = -15sinα/2
tanα = -2/15
|OA+OB|^2
=|(sinα+cosα,10)|^2
= (sinα+cosα)^2 +100
= 2sinαcosα + 101
= -2(2/√229)(15/√229) +101
= 23069/229
|OA+OB| = √(23069/229)
=OB - OA
=( cosα-sinα,8)
f(α)
=PB.CA
= -AB.CA
= -( cosα-sinα,8) . ( -2sinα,1)
=2sinαcosα-2(sinα)^2 + 8
= sin2α +(1-2(sinα)^2) + 7
= sin2α + cos2α + 7
= √2(sin(2α+π/4) + 7
值域 = [7-√2,7+√2]
增加 (-π/8,π/8]
减小[π/8, π/2)
(2)
let P be (x,y)
OPC三点共线
-2/sinα = y/x
-2x = ysinα
x = -ysinα/2
AB= BP
(cosα-sinα, 8) = (-ysinα/2-cosα, y-9)
=> y-9 = 8 and cosα-sinα= -ysinα/2-cosα
= y = 17 and cosα-sinα= -17sinα/2-cosα
cosα-sinα= -17sinα/2-cosα
2cosα = -15sinα/2
cosα = -15sinα/2
tanα = -2/15
|OA+OB|^2
=|(sinα+cosα,10)|^2
= (sinα+cosα)^2 +100
= 2sinαcosα + 101
= -2(2/√229)(15/√229) +101
= 23069/229
|OA+OB| = √(23069/229)
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