MT【261】左准线

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$(\overrightarrow{F_1A}+\overrightarrow{F_1B})\cdot\overrightarrow{AB}=0$, 求$l$的斜率_____

设$A,B$的中点为$M$,注意到$M$在左准线$x=-\dfrac{1}{2}$上,故设$M(-\dfrac{1}{2},m)$,
则$\overrightarrow{F_1A}\cdot\overrightarrow{F_1B}=(\dfrac{3}{2},m)\cdot(-\dfrac{5}{2},m)=m^2-\dfrac{15}{4}=0$
故$m=\pm\dfrac{\sqrt{15}}{2},$故$l$的斜率为$\pm\dfrac{\sqrt{15}}{2}$