已知A（2cosa,根号3sina）B（2cosb,根号3sinb）C（-1,0）是平面上三个不同的点且满足向量CA=m向量BC求m

已知A（2cosa,根号3sina）B（2cosb,根号3sinb）C（-1,0）是平面上三个不同的点且满足向量CA=m向量BC求m

CA=(-1-2cosa,-根号3 sina)
BC=(-1-2cosb,-根号3 sinb)
则-1-2cosa =(-1-2cosb)m ...(1)
-根号3 sina=-根号3 sinb *m ---(2)
sina=msinb ..(3)
-1-2cosa =-m-2mcosb
cosa =(m-1)/2 +mcosb ...(4)
sin^2 a+cos^2 a=m^2 sin^2 b +((m-1)/2+mcosb)^2 =1 ...(5)
m^2 -m^2 cos^2 b +(m-1)^2/4 +m(m-1)cosb +m^2 cos^2 b=1
m^2-1+(m-1)^2/4 +m(m-1)cosb=0
-1<=cosb =(1-m^2-(m-1)^2/4) /(m(m-1))<=1
由上式可以得到m

A=(2cosA, √3sinA) , B=(2cosB, √3sinB), C=(-1,0)

CA=mBC
(-1-2cosA, -√3sinA) = m(-1-2cosB, -√3sinB)
=>
1+2cosA= m(1+2cosB) (1) and
√3sinA = √3msinB (2)
from (2)
sinA= msinB
=> sinA/sinB = m
from (1)
m = (1+2cosA)/(1+2cosB)

By sine rule
a/sinA = b/sinB
sinA/sinB = a/b
= |BC|/|CA|
= { [(-1-2cosB)^2 + (-√3sinB)^2 ]/ [(-1-2cosA)^2+(-√3sinA)^2] }^(1/2)
= { [ (cosB)^2 + 4cosB + 4 ]/ [ (cosA)^2 + 4cosA + 4] } ^(1/2)
= (cosB+2)/(cosA+2)

m = (cosB+2)/(cosA+2) = (1+2cosA)/(1+2cosB)
=> (cosA+)(1+2cosA) = (cosB+2)(1+2cosB)
=> cosA= cosB
=> A=B

m = sinA/sinB =1追问取值范围复制去Google翻译翻译结果