设抛物线方程为y^2=2px(p>0)过焦点F的弦AB的倾斜角为α,求证:焦点弦长为AB=2p/sinα^2
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设抛物线方程为y^2=2px(p>0)过焦点F的弦AB的倾斜角为α,求证:焦点弦长为AB=2p/sinα^2
设抛物线方程为y^2=2px(p>0)过焦点F的弦AB的倾斜角为α,求证:焦点弦长为AB=2p/sinα^2
设抛物线方程为y^2=2px(p>0)过焦点F的弦AB的倾斜角为α,求证:焦点弦长为AB=2p/sinα^2
由抛物线方程y^2=2px,得抛物线的焦点坐标为(p/2,0),又AB的倾角为α,
∴AB的方程是:y=(x-p/2)tanα.
∴可设A、B的坐标分别是(m,(m-p/2)tanα)、(n,(n-p/2)tanα).
联立:y=(x-p/2)tanα、 y^2=2px,消去y,得:
[(x-p/2)tanα]^2=2px, ∴x^2(tanα)^2-px(tanα)^2+(ptanα)^2/4=2px,
∴4x^2(tanα)^2-[4p(tanα)^2+8p]x+p^2(tanα)^2=0.
显然,m、n是方程4x^2(tanα)^2-[4p(tanα)^2+8p]x+p^2(tanα)^2=0的两根,
∴由韦达定理,有:
m+n=[p(tanα)^2+2p]/(tanα)^2=p+2p/(tanα)^2、 mn=p^2/4.
而|AB|^2=(m-n)^2+[(m-p/2)tanα-(n-p/2)tanα]^2
=(m-n)^2+(m-n)^2(tanα)^2
=(m-n)^2[1+(tanα)^2]
=[(m+n)^2-4mn]/(cosα)^2
=[p^2+4p^2/(tanα)^2+4p^2/(tanα)^4-p^2]/(cosα)^2
=4p^2[1/(tanα)^2+1/(tanα)^4]/(cosα)^2
=4p^2[(tanα)^2+1]/[(tanα)^4(cosα)^2]
=4p^2[1/(cosα)^2]/[(sinα)^4/(cosα)^2]
=4p^2/(sinα)^4.
∴|AB|=4p/(sinα)^2.