已知P为双曲线x²/16-y²/b²=1(b>0)上的点,F1、F2为左右焦点,当向量PF1•向量PF2=0时,△PF1F2的面积为9,则b的值为
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![已知P为双曲线x²/16-y²/b²=1(b>0)上的点,F1、F2为左右焦点,当向量PF1•向量PF2=0时,△PF1F2的面积为9,则b的值为](/uploads/image/z/1670481-9-1.jpg?t=%E5%B7%B2%E7%9F%A5P%E4%B8%BA%E5%8F%8C%E6%9B%B2%E7%BA%BFx%26%23178%3B%2F16-y%26%23178%3B%2Fb%26%23178%3B%3D1%28b%3E0%29%E4%B8%8A%E7%9A%84%E7%82%B9%2CF1%E3%80%81F2%E4%B8%BA%E5%B7%A6%E5%8F%B3%E7%84%A6%E7%82%B9%2C%E5%BD%93%E5%90%91%E9%87%8FPF1%26%238226%3B%E5%90%91%E9%87%8FPF2%3D0%E6%97%B6%2C%E2%96%B3PF1F2%E7%9A%84%E9%9D%A2%E7%A7%AF%E4%B8%BA9%2C%E5%88%99b%E7%9A%84%E5%80%BC%E4%B8%BA)
已知P为双曲线x²/16-y²/b²=1(b>0)上的点,F1、F2为左右焦点,当向量PF1•向量PF2=0时,△PF1F2的面积为9,则b的值为
已知P为双曲线x²/16-y²/b²=1(b>0)上的点,F1、F2为左右焦点,
当向量PF1•向量PF2=0时,△PF1F2的面积为9,则b的值为
已知P为双曲线x²/16-y²/b²=1(b>0)上的点,F1、F2为左右焦点,当向量PF1•向量PF2=0时,△PF1F2的面积为9,则b的值为
∵向量PF1·向量PF2=0,∴PF1⊥PF2,∴(1/2)PF1×PF2=S(△PF1F2)=9,∴PF1×PF2=18.
由双曲线定义,有:|PF1-PF2|=2a=16,∴PF1^2+PF2^2-2PF1×PF2=256,
∴PF1^2+PF2^2=256+2×18=292.
由勾股定理,有:PF1^2+PF2^2=F1F2^2=(2c)^2=4(a^2+b^2)=4(16+b^2),
∴4(16+b^2)=292,∴16+b^2=64,∴b^2=48,∴b=4√3.