For a constant $a > 3$, two curves $y = a ^ { x - 1 }$ and $y = 3 ^ { x }$ meet at point P. Let the $x$-coordinate of point P be $k$. Let A be the point where the tangent line to the curve $y = 3 ^ { x }$ at point P meets the $x$-axis, and let B be the point where the tangent line to the curve $y = a ^ { x - 1 }$ at point P meets the $x$-axis. For point $\mathrm { H } ( k , 0 )$, when $\overline { \mathrm { AH } } = 2 \overline { \mathrm { BH } }$, what is the value of $a$? [4 points] (1) 6 (2) 7 (3) 8 (4) 9 (5) 10
For a constant $a > 3$, two curves $y = a ^ { x - 1 }$ and $y = 3 ^ { x }$ meet at point P. Let the $x$-coordinate of point P be $k$.
Let A be the point where the tangent line to the curve $y = 3 ^ { x }$ at point P meets the $x$-axis, and let B be the point where the tangent line to the curve $y = a ^ { x - 1 }$ at point P meets the $x$-axis. For point $\mathrm { H } ( k , 0 )$, when $\overline { \mathrm { AH } } = 2 \overline { \mathrm { BH } }$, what is the value of $a$? [4 points]\\
(1) 6\\
(2) 7\\
(3) 8\\
(4) 9\\
(5) 10