已知AB是抛物线y^2=2px(p>0)的焦点弦,为抛物线焦点,点A(X1,Y1),B(X2,Y2).求证:(1).Y1Y2=-P^2 X1X2=(P^2)/4(2).|AB|=X1+X2+P=2P/(SINa)^2(a为直线AB的倾斜角)
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![已知AB是抛物线y^2=2px(p>0)的焦点弦,为抛物线焦点,点A(X1,Y1),B(X2,Y2).求证:(1).Y1Y2=-P^2 X1X2=(P^2)/4(2).|AB|=X1+X2+P=2P/(SINa)^2(a为直线AB的倾斜角)](/uploads/image/z/3963635-35-5.jpg?t=%E5%B7%B2%E7%9F%A5AB%E6%98%AF%E6%8A%9B%E7%89%A9%E7%BA%BFy%5E2%3D2px%28p%3E0%29%E7%9A%84%E7%84%A6%E7%82%B9%E5%BC%A6%2C%E4%B8%BA%E6%8A%9B%E7%89%A9%E7%BA%BF%E7%84%A6%E7%82%B9%2C%E7%82%B9A%28X1%2CY1%29%2CB%28X2%2CY2%29.%E6%B1%82%E8%AF%81%EF%BC%9A%281%29.Y1Y2%3D-P%5E2+X1X2%3D%28P%5E2%29%2F4%282%29.%7CAB%7C%3DX1%2BX2%2BP%3D2P%2F%28SINa%29%5E2%28a%E4%B8%BA%E7%9B%B4%E7%BA%BFAB%E7%9A%84%E5%80%BE%E6%96%9C%E8%A7%92%EF%BC%89)
已知AB是抛物线y^2=2px(p>0)的焦点弦,为抛物线焦点,点A(X1,Y1),B(X2,Y2).求证:(1).Y1Y2=-P^2 X1X2=(P^2)/4(2).|AB|=X1+X2+P=2P/(SINa)^2(a为直线AB的倾斜角)
已知AB是抛物线y^2=2px(p>0)的焦点弦,为抛物线焦点,点A(X1,Y1),B(X2,Y2).求证:
(1).Y1Y2=-P^2 X1X2=(P^2)/4
(2).|AB|=X1+X2+P=2P/(SINa)^2(a为直线AB的倾斜角)
已知AB是抛物线y^2=2px(p>0)的焦点弦,为抛物线焦点,点A(X1,Y1),B(X2,Y2).求证:(1).Y1Y2=-P^2 X1X2=(P^2)/4(2).|AB|=X1+X2+P=2P/(SINa)^2(a为直线AB的倾斜角)
1.设直线AB的斜率为k (a为直线AB的倾斜角)
当a=π/2时,AB垂直于x轴,x=p/2
得y=±p
所以A B的坐标分别为(p/2,p),(p/2,-p)
y1*y2=-p^2,x1*x2=p^2/4
当a≠π/2
y^2=2px
焦点(p/2,0),准线x=-p/2
则直线AB:y=k(x-p/2)
抛物线:y^2=2px
联立
k^2x^2-(k^2p+2p)x+k^2*p^2/4=0
则x1*x2=p^2/4
y1*y2=-p^2
2.因为抛物线上任一点到焦点的距离等于其到准线的距离
所以|AB|=|AF|+|B=x1+P/2+x2+P/2=x1+x2+P
(1)当a=π/2时,AB垂直于x轴
令x=p/2得y=±p
所以A B的坐标分别为(p/2,p),(p/2,-p)
所以弦长AB=2p
(2)当a≠π/2时,焦点弦AB的的斜率为k=tana
所以直线为y-0=k(x-p/2)
带入抛物线y^2=2px(p>0)得k^2(x-p/2)^2=2px
化简得k^2x^2-(k^2p+2p)x+k^2p^2/4=0
所以x1+x2=(k^2p+2p)/k^2,x1*x2=p^2/4
所以弦长|AB|=√(1+k^2)*|x1-x2|=√(1+k^2)*√[(x1+x2)^2-4x1*x2]
=2(k^2+1)p/k^2=2p(tana^2+1)/tana^2 =2p/(sina)^2
按时地方