For the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( \mathrm { a } > 0 , \mathrm {~b} > 0 )$, let $F$ be the right focus, and $\mathrm { A } _ { 1 } , \mathrm {~A} _ { 2 }$ be the left and right vertices respectively. A line through $F$ perpendicular to $A _ { 1 } A _ { 2 }$ intersects the hyperbola at points $B$ and $C$. If $A _ { 1 } B \perp A _ { 2 } C$, then the slope of the asymptotes of the hyperbola is (A) $\pm \frac { 1 } { 2 }$ (B) $\pm \frac { \sqrt { 2 } } { 2 }$ (C) $\pm 1$ (D) $\pm \sqrt { 2 }$
For the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( \mathrm { a } > 0 , \mathrm {~b} > 0 )$, let $F$ be the right focus, and $\mathrm { A } _ { 1 } , \mathrm {~A} _ { 2 }$ be the left and right vertices respectively. A line through $F$ perpendicular to $A _ { 1 } A _ { 2 }$ intersects the hyperbola at points $B$ and $C$. If $A _ { 1 } B \perp A _ { 2 } C$, then the slope of the asymptotes of the hyperbola is
(A) $\pm \frac { 1 } { 2 }$
(B) $\pm \frac { \sqrt { 2 } } { 2 }$
(C) $\pm 1$
(D) $\pm \sqrt { 2 }$