过抛物线y^2=2px(p>0)的焦点F做倾斜角为α的直线与抛物线交于A,B两点,求证:|AB|=2p/(sinα)^2
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![过抛物线y^2=2px(p>0)的焦点F做倾斜角为α的直线与抛物线交于A,B两点,求证:|AB|=2p/(sinα)^2](/uploads/image/z/5534112-48-2.jpg?t=%E8%BF%87%E6%8A%9B%E7%89%A9%E7%BA%BFy%5E2%3D2px%EF%BC%88p%3E0%29%E7%9A%84%E7%84%A6%E7%82%B9F%E5%81%9A%E5%80%BE%E6%96%9C%E8%A7%92%E4%B8%BA%CE%B1%E7%9A%84%E7%9B%B4%E7%BA%BF%E4%B8%8E%E6%8A%9B%E7%89%A9%E7%BA%BF%E4%BA%A4%E4%BA%8EA%2CB%E4%B8%A4%E7%82%B9%2C%E6%B1%82%E8%AF%81%EF%BC%9A%EF%BD%9CAB%EF%BD%9C%3D2p%2F%28sin%CE%B1%29%5E2)
过抛物线y^2=2px(p>0)的焦点F做倾斜角为α的直线与抛物线交于A,B两点,求证:|AB|=2p/(sinα)^2
过抛物线y^2=2px(p>0)的焦点F做倾斜角为α的直线与抛物线交于A,B两点,求证:|AB|=2p/(sinα)^2
过抛物线y^2=2px(p>0)的焦点F做倾斜角为α的直线与抛物线交于A,B两点,求证:|AB|=2p/(sinα)^2
y^2=4x
焦点F(p/2,0)
准线x=-p/2
设焦点弦:y=tanα*(x-p/2) (α≠π/2)
y=tanα*(x-p/2)代入y^2=2px
(tanα)^2x^2-[(tanα)^2+2]px+(ptanα)^2/4=0
由根与系数关系
x1+x2=p[(tanα)^2+2]/(tanα)^2=[1+2/(tanα)^2]p
由抛物线上任意一点到焦点距离与到准线距离相等
|AB|=|AF|+|BF|
=|x1+p/2|+|x2+p/2|
=x1+x2+p
=2p[1+1/(tanα)]^2
=2p[1+(cotα)^2]
=2p(cscα)^2
=2p/(sinα)^2
当AP倾斜角为π/2时
|AB|=2p=2p/[sin(π/2)]^2
可知|AB|=2p/(sinα)^2
FA=FAcosa+p;(p就是表达式里的p,称焦参数,为焦点——准线的距离)
FB=FBcos(a+π)+p=-FBcosa+p
(左边为到焦点的距离,右边为到准线的距离)
所以
FA=p/(1+cosa)
FB=p/(1-cosa)
FA+FB=p(1+cosa+1-cosa)/[1-(cosa)^2]=2p/(sina)^2
说明:(该...
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FA=FAcosa+p;(p就是表达式里的p,称焦参数,为焦点——准线的距离)
FB=FBcos(a+π)+p=-FBcosa+p
(左边为到焦点的距离,右边为到准线的距离)
所以
FA=p/(1+cosa)
FB=p/(1-cosa)
FA+FB=p(1+cosa+1-cosa)/[1-(cosa)^2]=2p/(sina)^2
说明:(该方法未联立方程,至多到大学你会知道,它运用了极坐标系的思想)
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