双曲线弦长公式的推导(含ab的式子) 2ab^(1 k^)\|b^-ak^|

Question

如题所述

Answer 1

设焦点弦端点为A,B,A,B横坐标分别为x1,x2,A,B到与焦点对应的准线的距离分别为d1,d2,焦点弦过焦点F,则离心率e=AF/d1=BF/d2=(AF+BF)/(d1+d2)=AB/(d1+d2)=AB/[x1-(a^2)/c+x2-(a^2)/c]焦点弦长AB=e[x1-(a^2)/c+x2-(a^2)/c]若F为右焦点,则d1+d2=x1-(a^2)/c+x2-(a^2)/c=(a^2)/c-x1+(a^2)/c-x2=2(a^2)/c-(x1+x2)焦点弦长AB=e[x1-(a^2)/c+x2-(a^2)/c]=e[2(a^2)/c-(x1+x2)]=2(c/a)(a^2)/c-e(x1+x2)=2a-e(x1+x2)若F为左焦点,则d1+d2=x1-(a^2)/c+x2-(a^2)/c=x1-(a^2)/c+x2-(a^2)/c=(x1+x2)-2(a^2)/c焦点弦长AB=e[x1-(a^2)/c+x2-(a^2)/c]=e[(x1+x2)-2(a^2)/c]=e(x1+x2)-2(c/a)(a^2)/c=e(x1+x2)-2a