9. Let $A = \{(x,y) \mid x^2 + y^2 \leq 1, x, y \in \mathbf{Z}\}$, $B = \{(x,y) \mid |x| \leq 2, |y| \leq 2, x, y \in \mathbf{Z}\}$. Define $A \oplus B = \{(x_1 + x_2, y_1 + y_2) \mid (x_1, y_1) \in A, (x_2, y_2) \in B\}$. The number of elements in $A \oplus B$ is
A. 77
B. 49
C. 45
D. 30

[Optional 4-4: Coordinate Systems and Parametric Equations] (10 points)
In the rectangular coordinate system $xOy$, the parametric equation of curve $C$ is $\left\{\begin{array}{l} x = 3\cos\theta \\ y = \sin\theta \end{array}\right.$ ($\theta$ is the parameter), and the parametric equation of line $l$ is $\left\{\begin{array}{l} x = a + 4t \\ y = 1 - t \end{array}\right.$ ($t$ is the parameter).
(1) If $a = -1$, find the coordinates of the intersection points of $C$ and $l$.
(2) If the maximum distance from a point on $C$ to line $l$ is $\sqrt{17}$, find $a$.

[Elective 4-4: Polar Coordinates and Parametric Equations]
In the rectangular coordinate system $x O y$, the parametric equation of curve $C$ is $\left\{ \begin{array} { l } x = 2 \cos \theta , \\ y = 4 \sin \theta \end{array} \right.$ ($\theta$ is the parameter). The parametric equation of line $l$ is $\left\{ \begin{array} { l } x = 1 + t \cos \alpha , \\ y = 2 + t \sin \alpha \end{array} \right.$ ($t$ is the parameter).
(1) Find the rectangular coordinate equations of $C$ and $l$;
(2) If the midpoint of the line segment obtained by the intersection of curve $C$ and line $l$ is $( 1,2 )$, find the slope of $l$.

[Elective 4-4: Coordinate Systems and Parametric Equations] In the rectangular coordinate system $x O y$, the parametric equation of curve $C _ { 1 }$ is $\left\{ \begin{array} { l } x = \frac { 2 + t } { 6 } \\ y = \sqrt { t } \end{array} \right.$ ($t$ is the parameter), and the parametric equation of curve $C _ { 2 }$ is $\left\{ \begin{array} { l } x = - \frac { 2 + s } { 6 } \\ y = - \sqrt { s } \end{array} \right.$ ($s$ is the parameter).
(1) Write the ordinary equation of $C _ { 1 }$ ;
(2) With the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of curve $C _ { 3 }$ is $2 \cos \theta - \sin \theta = 0$ . Find the rectangular coordinates of the intersection points of $C _ { 3 }$ with $C _ { 1 }$, and the rectangular coordinates of the intersection points of $C _ { 3 }$ with $C _ { 2 }$ .

[Elective 4-4: Coordinate Systems and Parametric Equations] (10 points) In the rectangular coordinate system $xOy$, the parametric equation of curve $C _ { 1 }$ is $\left\{ \begin{array} { l } x = \frac { 2 + t } { 6 } \\ y = \sqrt { t } \end{array} \right.$ ($t$ is the parameter), and the parametric equation of curve $C _ { 2 }$ is $\left\{ \begin{array} { l } x = - \frac { 2 + s } { 6 } \\ y = - \sqrt { s } \end{array} \right.$ ($s$ is the parameter).
(1) Write the Cartesian equation of $C _ { 1 }$;
(2) With the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of curve $C _ { 3 }$ is $2 \cos \theta - \sin \theta = 0$. Find the rectangular coordinates of the intersection points of $C _ { 3 }$ with $C _ { 1 }$, and the rectangular coordinates of the intersection points of $C _ { 3 }$ with $C _ { 2 }$.

We denote $\mathscr{C}$ the image in $\mathbb{R}^2$ of the application $$\gamma : \left[\frac{\pi}{4}, \frac{7\pi}{4}\right] \rightarrow \mathbb{R}^2, t \mapsto (\cos t, 2\sin t)$$
III.A.1) Represent $\mathscr{C}$. III.A.2) Specify the following topological properties of $\mathscr{C}$. a) Is it an open set of $\mathbb{R}^2$? b) A closed set? c) A bounded set? d) A compact? e) A convex set?

Let $x = 2 t , y = \frac { t ^ { 2 } } { 3 }$ be a conic. Let $S$ be the focus and $B$ be the point on the axis of the conic such that $S A \perp B A$, where $A$ is any point on the conic. If $k$ is the ordinate of the centroid of the $\triangle S A B$, then $\lim _ { t \rightarrow 1 } k$ is equal to
(1) $\frac { 17 } { 18 }$
(2) $\frac { 19 } { 18 }$
(3) $\frac { 11 } { 18 }$
(4) $\frac { 13 } { 18 }$

The ordinates of the points $P$ and $Q$ on the parabola with focus $( 3,0 )$ and directrix $x = - 3$ are in the ratio 3:1. If $R ( \alpha , \beta )$ is the point of intersection of the tangents to the parabola at $P$ and $Q$, then $\frac { \beta ^ { 2 } } { \alpha }$ is equal to

Let PQ be a chord of the parabola $y ^ { 2 } = 12 x$ and the midpoint of PQ be at $( 4,1 )$. Then, which of the following point lies on the line passing through the points P and Q ?
(1) $( 3 , - 3 )$
(2) $( 2 , - 9 )$
(3) $\left( \frac { 3 } { 2 } , - 16 \right)$
(4) $\left( \frac { 1 } { 2 } , - 20 \right)$

A rod of length eight units moves such that its ends $A$ and $B$ always lie on the lines $x - y + 2 = 0$ and $y + 2 = 0$, respectively. If the locus of the point $P$, that divides the rod $AB$ internally in the ratio $2 : 1$ is $9 \left( x ^ { 2 } + \alpha y ^ { 2 } + \beta x y + \gamma x + 28 y \right) - 76 = 0$, then $\alpha - \beta - \gamma$ is equal to :
(1) 22
(2) 21
(3) 23
(4) 24

On a square piece of paper, there is a point $P$ that is 6 cm from the left edge and 8 cm from the bottom edge. Now fold the bottom-left corner $O$ of the paper inward to point $P$, as shown in the figure. The area of the folded triangle is $\square$ square centimeters.