过抛物线y^2=2Px(p>0)的焦点F作倾斜角为π/4的直线,交抛物线于A,B两点,点A在x轴的上方,求|AF|/|BF|的值
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![过抛物线y^2=2Px(p>0)的焦点F作倾斜角为π/4的直线,交抛物线于A,B两点,点A在x轴的上方,求|AF|/|BF|的值](/uploads/image/z/5534117-53-7.jpg?t=%E8%BF%87%E6%8A%9B%E7%89%A9%E7%BA%BFy%5E2%3D2Px%28p%3E0%29%E7%9A%84%E7%84%A6%E7%82%B9F%E4%BD%9C%E5%80%BE%E6%96%9C%E8%A7%92%E4%B8%BA%CF%80%2F4%E7%9A%84%E7%9B%B4%E7%BA%BF%2C%E4%BA%A4%E6%8A%9B%E7%89%A9%E7%BA%BF%E4%BA%8EA%2CB%E4%B8%A4%E7%82%B9%2C%E7%82%B9A%E5%9C%A8x%E8%BD%B4%E7%9A%84%E4%B8%8A%E6%96%B9%2C%E6%B1%82%EF%BD%9CAF%EF%BD%9C%2F%EF%BD%9CBF%EF%BD%9C%E7%9A%84%E5%80%BC)
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过抛物线y^2=2Px(p>0)的焦点F作倾斜角为π/4的直线,交抛物线于A,B两点,点A在x轴的上方,求|AF|/|BF|的值
过抛物线y^2=2Px(p>0)的焦点F作倾斜角为π/4的直线,交抛物线于A,B两点,点A在x轴的上方,求|AF|/|BF|的值
过抛物线y^2=2Px(p>0)的焦点F作倾斜角为π/4的直线,交抛物线于A,B两点,点A在x轴的上方,求|AF|/|BF|的值
直线为y=x-p/2,联立y=x-p/2,y^2=2Px解得xA=3p/2+√2p,xB=3p/2-√2p
|AF|/|BF|=(xA+p/2)/(xB+p/2)=(2p+√2p)/(2p-√2p)=3+2√2
du(短)=|BF|
ch(长)=|AF|
因为斜率为1(斜边为直角边的根号2倍),又是抛物线(到焦点距离等于到准线距离),就有
ch = |AF| = A到准线距离 = B到准线距离 + (du+ch)/ 根号2
马上就得到:ch*(1-√2/2) = du*(1+√2/2)
ch / du = 3 + 2√2