[Calculus] The tangent line to the curve $y = e ^ { x }$ at the point $( 1 , e )$ is tangent to the curve $y = 2 \sqrt { x - k }$. What is the value of the real number $k$? [3 points]
(1) $\frac { 1 } { e }$
(2) $\frac { 1 } { e ^ { 2 } }$
(3) $\frac { 1 } { e ^ { 4 } }$
(4) $\frac { 1 } { 1 + e }$
(5) $\frac { 1 } { 1 + e ^ { 2 } }$

If line $l$ is tangent to both the curve $y = \sqrt { x }$ and the circle $x ^ { 2 } + y ^ { 2 } = \frac { 1 } { 5 }$ , then the equation of $l$ is
A. $y = 2 x + 1$
B. $y = 2 x + \frac { 1 } { 2 }$
C. $y = \frac { 1 } { 2 } x + 1$
D. $y = \frac { 1 } { 2 } x + \frac { 1 } { 2 }$

If the tangent line to the curve $y = \mathrm { e } ^ { x } + x$ at the point $( 0,1 )$ is also a tangent line to the curve $y = \ln ( x + 1 ) + a$ , then $a = $ $\_\_\_\_$ .

The slope of the line touching both the parabolas $y ^ { 2 } = 4 x$ and $x ^ { 2 } = - 32 y$ is
(1) $\frac { 1 } { 8 }$
(2) $\frac { 2 } { 3 }$
(3) $\frac { 1 } { 2 }$
(4) $\frac { 3 } { 2 }$

In the rectangular coordinate plane, the tangent line drawn to the graph of the function $f ( x ) = x ^ { 2 } + a x$ at the point $( 2 , f ( 2 ) )$ is tangent to the graph of the function $g ( x ) = b x ^ { 3 }$ at the point $( 1 , g ( 1 ) )$. Accordingly, what is the product $\mathbf { a } \cdot \mathbf { b }$?
A) 2
B) 4
C) 6
D) 8
E) 10