$$y = x ^ { 2 } - 2 ( a + 1 ) x + a ^ { 2 } - 1$$
If the parabola is tangent to the line $y = 1$, what is a?
A) $\frac { -3 } { 2 }$
B) $\frac { -3 } { 4 }$
C) 0
D) 1
E) 2

$$\sqrt { x } - \sqrt { y } = \sqrt { x + y - 1 }$$
Given this, what is the product $x \cdot y$?
A) $\frac { 1 } { 3 }$
B) $\frac { 1 } { 4 }$
C) $\frac { 3 } { 4 }$
D) $\frac { 2 } { 5 }$
E) $\frac { 4 } { 5 }$

The parabolas $f ( x )$ and $g ( x )$ whose graphs are shown above intersect each other at their vertices.
Given this, what is the value of $\mathbf { g } ( \mathbf { 0 } )$?
A) 3
B) 4
C) 5
D) 6
E) 7

Let $a$ and $b$ be positive real numbers. In the Cartesian coordinate plane, using the parabola
$$p ( x ) = ( x - a ) ^ { 2 } - b$$
that passes through the origin, three parabolas defined as
$$\begin{aligned} & p ( x + a ) + b \\ & p ( x + a ) - b \\ & p ( x - a ) - b \end{aligned}$$
have their vertices at the vertices of a triangle with an area of 16 square units.
Accordingly, what is the sum $a + b$?
A) 6 B) 9 C) 12 D) 15 E) 18

Let $a$ and $b$ be real numbers. In the rectangular coordinate plane, the parabola $y = x^{2} + ax + b$ is tangent to the $x$-axis and to the line $y = x$.
What is the product $a \cdot b$?
A) $\dfrac{1}{2}$ B) $\dfrac{1}{4}$ C) $\dfrac{1}{8}$ D) $\dfrac{1}{16}$ E) $\dfrac{1}{32}$