On the coordinate plane, there is a figure $\Gamma$ with equation $(x-1)^2 + (y-1)^2 = 101$. Select the correct options.
(1) $\Gamma$ intersects the negative $x$-axis and negative $y$-axis at $(-9, 0)$ and $(0, -9)$ respectively
(2) The point on $\Gamma$ with the maximum $x$-coordinate is $(11, 0)$
(3) The maximum distance from a point on $\Gamma$ to the origin is $\sqrt{2} + \sqrt{101}$
(4) Points on $\Gamma$ in the third quadrant can be expressed in polar coordinates as $[9, \theta]$, where $\pi < \theta < \frac{3}{2}\pi$
(5) After a rotational linear transformation, the figure can still be expressed by a quadratic equation in two variables without an $xy$ term

Suppose $A, B$ are two points on a parabola $\Gamma$ and their connecting line segment passes through the focus $F$ of $\Gamma$. Let the projections of $A, F, B$ onto the directrix of $\Gamma$ be $A', F', B'$ respectively. Select the option equal to $\frac{\overline{AF}}{\overline{A'F'}}$. (Note: This schematic diagram only illustrates the relative positions of the points; the distance relationships between points are not accurate)
(1) $\tan \angle 1$, where $\angle 1 = \angle A'F'A$
(2) $\sin \angle 2$, where $\angle 2 = \angle AF'F$
(3) $\sin \angle 3$, where $\angle 3 = \angle A'AF$
(4) $\cos \angle 4$, where $\angle 4 = \angle F'FB$
(5) $\tan \angle 5$, where $\angle 5 = \angle FF'B$

There is a shooting game with the launcher placed at the origin of a coordinate plane and three circular target disks with radius 1, centered at $(2,2)$, $(4,6)$, and $(8,1)$ respectively. A player selects a positive number $a$ and presses a button. The launcher then fires a laser beam in the direction of point $(1, a)$ (forming a ray). Assume the laser beam can penetrate through the target after hitting it and continue in the original direction (grazing the edge of the disk also counts as a hit). Select the correct options.
(1) The laser beam lies on a line passing through the origin with slope $a$
(2) If $a = \frac{3}{2}$, the laser beam will hit the disk centered at $(4,6)$
(3) The player can hit all three disks with just one laser beam
(4) The player needs to fire at least three laser beams to hit all three disks
(5) If the player fires one laser beam and hits the disk centered at $(8,1)$, then $a \leq \frac{16}{63}$

On the coordinate plane, there is an annular region formed by the intersection of the exterior of the circle $x ^ { 2 } + y ^ { 2 } = 3$ and the interior of the circle $x ^ { 2 } + y ^ { 2 } = 4$ . A person wants to use a straight scanning rod of length 1 to scan a certain region $R$ above the $x$-axis of this annular region. He designs the scanning rod with black and white ends moving respectively on the semicircles $C _ { 1 } : x ^ { 2 } + y ^ { 2 } = 3 ( y \geq 0 )$ and $C _ { 2 } : x ^ { 2 } + y ^ { 2 } = 4 ( y \geq 0 )$ . Initially, the black end of the scanning rod is at point $A ( \sqrt { 3 } , 0 )$ and the white end is at point $B$ on $C _ { 2 }$ . Then the black and white ends move counterclockwise along $C _ { 1 }$ and $C _ { 2 }$ respectively until the white end reaches point $B ^ { \prime } ( - 2,0 )$ on $C _ { 2 }$ , at which point scanning stops.
Let $O$ be the origin. When the scanning rod stops, the positions of the black and white ends are $A ^ { \prime }$ and $B ^ { \prime }$ respectively. In the diagram area of the answer sheet, use hatching to indicate the region $R$ swept by the scanning rod; and in the solution area, find $\cos \angle O A ^ { \prime } B ^ { \prime }$ and the polar coordinates of point $A ^ { \prime }$ . (Non-multiple choice question, 6 points)

On the coordinate plane, there is an annular region formed by the intersection of the exterior of the circle $x ^ { 2 } + y ^ { 2 } = 3$ and the interior of the circle $x ^ { 2 } + y ^ { 2 } = 4$ . A person wants to use a straight scanning rod of length 1 to scan a certain region $R$ above the $x$-axis of this annular region. He designs the scanning rod with black and white ends moving respectively on the semicircles $C _ { 1 } : x ^ { 2 } + y ^ { 2 } = 3 ( y \geq 0 )$ and $C _ { 2 } : x ^ { 2 } + y ^ { 2 } = 4 ( y \geq 0 )$ . Initially, the black end of the scanning rod is at point $A ( \sqrt { 3 } , 0 )$ and the white end is at point $B$ on $C _ { 2 }$ . Then the black and white ends move counterclockwise along $C _ { 1 }$ and $C _ { 2 }$ respectively until the white end reaches point $B ^ { \prime } ( - 2,0 )$ on $C _ { 2 }$ , at which point scanning stops.
(Continuing from Question 19) Let $\Omega$ denote the region swept by the scanning rod in the first quadrant. Find the areas of $\Omega$ and $R$ respectively. (Non-multiple choice question, 6 points)

On a coordinate plane, there is a circle with center $A(a, b)$ that is tangent to both coordinate axes. There is also a point $P(c, c)$, where $a > c > 0$, and it is known that $\overline{PA} = a + c$. Select the correct options.
(1) $a = b$ (2) Point $P$ is on the line $x + y = 0$ (3) Point $P$ is inside the circle (4) $\frac{a + c}{b - c} = \sqrt{2}$ (5) $\frac{a}{c} = 2 + 3\sqrt{2}$

Let $a , b$ be real numbers, and let $O$ be the origin of the coordinate plane. It is known that the graph of the quadratic function $f ( x ) = a x ^ { 2 }$ and the circle $\Omega : x ^ { 2 } + y ^ { 2 } - 3 y + b = 0$ both pass through point $P \left( 1 , \frac { 1 } { 2 } \right)$, and let point $C$ be the center of $\Omega$.
Find the cosine of the angle between vectors $\overrightarrow { C O }$ and $\overrightarrow { C P }$.

Let $a , b$ be real numbers, and let $O$ be the origin of the coordinate plane. It is known that the graph of the quadratic function $f ( x ) = a x ^ { 2 }$ and the circle $\Omega : x ^ { 2 } + y ^ { 2 } - 3 y + b = 0$ both pass through point $P \left( 1 , \frac { 1 } { 2 } \right)$, and let point $C$ be the center of $\Omega$.
Prove that the graph of $y = f ( x )$ and $\Omega$ have a common tangent line at point $P$.

On the coordinate plane, the equation of ellipse $\Gamma$ is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{6^{2}} = 1$ (where $a$ is a positive real number). If $\Gamma$ is scaled by a factor of 2 in the $x$-axis direction and by a factor of 3 in the $y$-axis direction with the origin $O$ as the center, the resulting new figure passes through the point $(18, 0)$. Which of the following options is a focus of $\Gamma$?
(1) $(0, 3\sqrt{3})$
(2) $(-3\sqrt{5}, 0)$
(3) $(0, 6\sqrt{13})$
(4) $(-3\sqrt{13}, 0)$
(5) $(9, 0)$

On the coordinate plane, let $\Gamma$ be a circle with center at the origin, and $P$ be one of the intersection points of $\Gamma$ and the $x$-axis. It is known that the line passing through $P$ with slope $\frac{1}{2}$ intersects $\Gamma$ at another point $Q$, and $\overline{PQ} = 1$. Then the radius of $\Gamma$ is . (Express as a simplified radical)