For a constant $a$ ($a > 1$), let A be a point in the first quadrant on the curve $y = a ^ { x } - 2$. Let B be the point where the line passing through A and parallel to the $y$-axis meets the $x$-axis, and let C be the point where this line meets the asymptote of the curve $y = a ^ { x } - 2$. If $\overline { \mathrm { AB } } = \overline { \mathrm { BC } }$ and the area of triangle AOC is 8, what is the value of $a \times \overline { \mathrm { OB } }$? (Here, O is the origin.) [4 points] (1) $2 ^ { \frac { 13 } { 6 } }$ (2) $2 ^ { \frac { 7 } { 3 } }$ (3) $2 ^ { \frac { 5 } { 2 } }$ (4) $2 ^ { \frac { 8 } { 3 } }$ (5) $2 ^ { \frac { 17 } { 6 } }$
For a constant $a$ ($a > 1$), let A be a point in the first quadrant on the curve $y = a ^ { x } - 2$. Let B be the point where the line passing through A and parallel to the $y$-axis meets the $x$-axis, and let C be the point where this line meets the asymptote of the curve $y = a ^ { x } - 2$. If $\overline { \mathrm { AB } } = \overline { \mathrm { BC } }$ and the area of triangle AOC is 8, what is the value of $a \times \overline { \mathrm { OB } }$? (Here, O is the origin.) [4 points]\\
(1) $2 ^ { \frac { 13 } { 6 } }$\\
(2) $2 ^ { \frac { 7 } { 3 } }$\\
(3) $2 ^ { \frac { 5 } { 2 } }$\\
(4) $2 ^ { \frac { 8 } { 3 } }$\\
(5) $2 ^ { \frac { 17 } { 6 } }$