[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 $l$.

22. Solution: (1) From the problem conditions, $|a| = 1$. Thus the parametric equation of $l$ is $\left\{\begin{array}{l} x = 4t + 1 \\ y = 3t - 1 \end{array}\right.$ ($t$ is the parameter). The parametric equation of circle $C$ is $\left\{\begin{array}{l} x = 1 + \cos\theta \\ y = -2 + \sin\theta \end{array}\right.$ ($\theta$ is the parameter). & \hline \end{tabular}
Eliminating parameter $t$, the ordinary equation of $l$ is $3x - 4y - 7 = 0$. ..... 3 marks
Eliminating parameter $\theta$, the ordinary equation of $C$ is $(x-1)^2 + (y+2)^2 = 1$. ..... 5 marks
(2) The equation of $l'$ is $y = \frac{3}{4}(x + m) - \frac{7}{4}$, i.e., $3x - 4y + 3m - 7 = 0$. ..... 6 marks
Since circle $C$ has only one point at distance $\mathbf{1}$ from $l'$, and the radius of circle $C$ is $\mathbf{1}$, the distance from $C(1, -2)$ to $l'$ is 2. ..... 8 marks
That is, $\frac{|3 + 8 + 3m - 7|}{5} = 2$. Solving, we get $m = 2$ ($m = -\frac{14}{3} < 0$ is rejected). ..... 10 marks

22. Solution: (1) Since $-1 < \frac{1-t^2}{1+t^2} \leq 1$ and $x^2 + \left(\frac{y}{2}\right)^2 = \left(\frac{1-t^2}{1+t^2}\right)^2 + \frac{4t^2}{(1+t^2)^2} = 1$, the rectangular coordinate equation of $C$ is $x^2 + \frac{y^2}{4} = 1$ $(x \neq -1)$. The rectangular coordinate equation of $l$ is $2x + \sqrt{3}y + 11 = 0$.
(2) From (1) we can set the parametric equation of $C$ as $\left\{\begin{array}{l} x = \cos\alpha, \\ y = 2\sin\alpha \end{array}\right.$ ($\alpha$ is the parameter, $-\pi < \alpha < \pi$). The distance from a point on $C$ to $l$ is $\frac{|2\cos\alpha + 2\sqrt{3}\sin\alpha + 11|}{\sqrt{7}} = \frac{4\cos\left(\alpha - \frac{\pi}{3}\right) + 11}{\sqrt{7}}$.
When $\alpha = -\frac{2\pi}{3}$, $4\cos\left(\alpha - \frac{\pi}{3}\right) + 11$ attains its minimum value of 7, therefore the minimum distance from a point on $C$ to $l$ is $\sqrt{7}$.

[Elective 4-4: Coordinate Systems and Parametric Equations] (10 points)
In the rectangular coordinate system $x O y$, the parametric equation of curve $C _ { 1 }$ is $\left\{ \begin{array} { l } x = \cos ^ { k } t , \\ y = \sin ^ { k } t \end{array} \right.$ ($t$ is the parameter). Establishing a polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of curve $C _ { 2 }$ is $$4 \rho \cos \theta - 16 \rho \sin \theta + 3 = 0$$
(1) When $k = 1$ , what type of curve is $C _ { 1 }$?
(2) When $k = 4$ , find the rectangular coordinates of the common points of $C _ { 1 }$ and $C _ { 2 }$ .

[Elective 4-4: Coordinate Systems and Parametric Equations] In the rectangular coordinate system $x O y$, the parametric equation of curve $C$ is $\left\{ \begin{array} { l } x = 2 - t - t ^ { 2 } , \\ y = 2 - 3 t + t ^ { 2 } \end{array} ( t \right.$ is a parameter and $t \neq 1 )$. $C$ intersects the coordinate axes at points $A , B$.
(1) Find $| A B |$;
(2) With the origin as the pole and the positive $x$-axis as the polar axis, establish a polar coordinate system and find the polar equation of line $A B$ .

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) $\frac { 13 } { 18 }$

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