1.若α∈(π/2,π),且3cos2α=sin(π/4-α),则sin2α=?2.过双曲线x²/a²-y²/b²=1(a>0,b>0)的右焦点F2作斜率为-1的直线,该直线与双曲线的两条渐近线的焦点分别为A,B,若向量F2A=向量AB,
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![1.若α∈(π/2,π),且3cos2α=sin(π/4-α),则sin2α=?2.过双曲线x²/a²-y²/b²=1(a>0,b>0)的右焦点F2作斜率为-1的直线,该直线与双曲线的两条渐近线的焦点分别为A,B,若向量F2A=向量AB,](/uploads/image/z/8573806-46-6.jpg?t=1.%E8%8B%A5%CE%B1%E2%88%88%EF%BC%88%CF%80%2F2%2C%CF%80%EF%BC%89%2C%E4%B8%943cos2%CE%B1%3Dsin%EF%BC%88%CF%80%2F4-%CE%B1%EF%BC%89%2C%E5%88%99sin2%CE%B1%3D%3F2.%E8%BF%87%E5%8F%8C%E6%9B%B2%E7%BA%BFx%26%23178%3B%2Fa%26%23178%3B-y%26%23178%3B%2Fb%26%23178%3B%3D1%EF%BC%88a%EF%BC%9E0%2Cb%EF%BC%9E0%EF%BC%89%E7%9A%84%E5%8F%B3%E7%84%A6%E7%82%B9F2%E4%BD%9C%E6%96%9C%E7%8E%87%E4%B8%BA-1%E7%9A%84%E7%9B%B4%E7%BA%BF%2C%E8%AF%A5%E7%9B%B4%E7%BA%BF%E4%B8%8E%E5%8F%8C%E6%9B%B2%E7%BA%BF%E7%9A%84%E4%B8%A4%E6%9D%A1%E6%B8%90%E8%BF%91%E7%BA%BF%E7%9A%84%E7%84%A6%E7%82%B9%E5%88%86%E5%88%AB%E4%B8%BAA%2CB%2C%E8%8B%A5%E5%90%91%E9%87%8FF2A%3D%E5%90%91%E9%87%8FAB%2C)
1.若α∈(π/2,π),且3cos2α=sin(π/4-α),则sin2α=?2.过双曲线x²/a²-y²/b²=1(a>0,b>0)的右焦点F2作斜率为-1的直线,该直线与双曲线的两条渐近线的焦点分别为A,B,若向量F2A=向量AB,
1.若α∈(π/2,π),且3cos2α=sin(π/4-α),则sin2α=?
2.过双曲线x²/a²-y²/b²=1(a>0,b>0)的右焦点F2作斜率为-1的直线,该直线与双曲线的两条渐近线的焦点分别为A,B,若向量F2A=向量AB,则双曲线的渐近线为?
3.设|AB|=1,若|CA|=2|CB|,则向量CA*向量CB的最大值为?
1.若α∈(π/2,π),且3cos2α=sin(π/4-α),则sin2α=?2.过双曲线x²/a²-y²/b²=1(a>0,b>0)的右焦点F2作斜率为-1的直线,该直线与双曲线的两条渐近线的焦点分别为A,B,若向量F2A=向量AB,
1.
3cos2α=sin(π/4-α) ==> 3(cosα+sinα)(cosα-sinα)=√2/2(cosα-sinα)
==> cosα+sinα=√2/6 ==> cos²α+sin²α+sin2α=1/18 ==>sin2α=-17/18
2
过右焦点F2斜率为-1的直线:y=c-x
y=c-x 与y=-b/ax交于B(ac/(a-b),bc/(b-a))
∵向量F2A=向量AB,∴A是线段F2B的中点
∴ F1B//OA ∴[bc/(b-a)]/[ac/(a-b)+c]=b/a
∴b/(b-2a)=b/a ==>b=3a==>b/a=3
∴渐近线:y=±3x
3
∵1= |AB|≥|CA|-|CB|=|CB|,1=|AB|≤|CA|+|CB|=3|CB|
∴1/3≤|CB|≤1,|CA|≤2 ∴向量CA*向量CB≤|CA||CB|=2
即向量CA*向量CB的最大值为2