Consider three distinct points $A$, $B$, $C$ in the coordinate plane, where point $A$ is $(1, 1)$. Circles are drawn with line segments $\overline{AB}$ and $\overline{AC}$ as diameters. These two circles intersect at point $A$ and point $P(4, 2)$. Given that $\overline{PB} = 3\sqrt{10}$ and point $B$ is in the fourth quadrant, the coordinates of point $B$ are (（12），（13）（14）).

On the coordinate plane, a circle with radius 12 intersects the line $x + y = 0$ at two points, and the distance between these two points is 8. If this circle intersects the line $x + y = 24$ at points $P$ and $Q$, then the length of segment $\overline { P Q }$ is $\_\_\_\_$ (14)$\sqrt { (15) }$. (Express as a simplified radical)

On the coordinate plane, there is a regular hexagon $A B C D E F$ with side length 3, where $A ( 3,0 ) , D ( - 3,0 )$. How many intersection points does the ellipse $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 7 } = 1$ have with the regular hexagon $A B C D E F$?
(1) 0
(2) 2
(3) 4
(4) 6
(5) 8

As shown in the figure, $L$ is a line passing through the origin $O$ on the coordinate plane, $\Gamma$ is a circle centered at $O$, and $L$ and $\Gamma$ have one intersection point $A ( 3,4 )$. It is known that $B , C$ are two distinct points on $\Gamma$ satisfying $\overrightarrow { B C } = \overrightarrow { O A }$. Select the correct options.
(1) The other intersection point of $L$ and $\Gamma$ is $( - 4 , - 3 )$
(2) The slope of line $B C$ is $\frac { 3 } { 4 }$
(3) $\angle A O C = 60 ^ { \circ }$
(4) The area of $\triangle A B C$ is $\frac { 25 \sqrt { 3 } } { 2 }$
(5) $B$ and $C$ are in the same quadrant

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

On a coordinate plane, there is a circle with radius 7 and center $O$. It is known that points $A, B$ are on the circle and $\overline{AB} = 8$. Then the dot product $\overrightarrow{OA} \cdot \overrightarrow{OB} =$ (14--1) (14--2).

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.
What are the coordinates of point $B$? (Single-choice question, 3 points)
(1) $( 0,2 )$
(2) $( 1 , \sqrt { 3 } )$
(3) $( \sqrt { 2 } , \sqrt { 2 } )$
(4) $( \sqrt { 3 } , 1 )$
(5) $( 2,0 )$

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) $\frac{a}{c} = 2 + 3\sqrt{2}$

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)