设双曲线$x^2-dfrac{y^2}{3}=1$的左右焦点为$F_1,F_2$, 直线$l$ 过$F_2$且与双曲线交于$A,B$两点.若$l$的斜率存在,且$(overrightarrow{F_1A}+overrightarrow{F_1B})cdotoverrightarrow{AB}=0$, 求$l$的斜率_____
设$A,B$的中点为$M$,因为$(overrightarrow{F_1A}+overrightarrow{F_1B})cdotoverrightarrow{AB}=0$得$F_1A=F_1B$,由第二定义，A，B两点到左准线距离相等.
所以$M$在左准线$x=-dfrac{1}{2}$上,故设$M(-dfrac{1}{2},m)$,
则$overrightarrow{F_1M}cdotoverrightarrow{AB}=(dfrac{3}{2},m)cdot(-dfrac{5}{2},m)=m^2-dfrac{15}{4}=0$
故$m=pmdfrac{sqrt{15}}{2},$故$l$的斜率为$pmdfrac{sqrt{15}}{2}$