Let region $A$ be enclosed by the curve $y = e ^ { 2 x }$, the $y$-axis, and the line $y = - 2 x + a$, and let region $B$ be enclosed by the curve $y = e ^ { 2 x }$ and the two lines $y = - 2 x + a , x = 1$. When the area of $A$ equals the area of $B$, what is the value of the constant $a$? (Here, $1 < a < e ^ { 2 }$) [3 points] (1) $\frac { e ^ { 2 } + 1 } { 2 }$ (2) $\frac { 2 e ^ { 2 } + 1 } { 4 }$ (3) $\frac { e ^ { 2 } } { 2 }$ (4) $\frac { 2 e ^ { 2 } - 1 } { 4 }$ (5) $\frac { e ^ { 2 } - 1 } { 2 }$
Let region $A$ be enclosed by the curve $y = e ^ { 2 x }$, the $y$-axis, and the line $y = - 2 x + a$, and let region $B$ be enclosed by the curve $y = e ^ { 2 x }$ and the two lines $y = - 2 x + a , x = 1$. When the area of $A$ equals the area of $B$, what is the value of the constant $a$? (Here, $1 < a < e ^ { 2 }$) [3 points]\\
(1) $\frac { e ^ { 2 } + 1 } { 2 }$\\
(2) $\frac { 2 e ^ { 2 } + 1 } { 4 }$\\
(3) $\frac { e ^ { 2 } } { 2 }$\\
(4) $\frac { 2 e ^ { 2 } - 1 } { 4 }$\\
(5) $\frac { e ^ { 2 } - 1 } { 2 }$