10. Given that the foci of ellipse $C$ are $F _ { 1 } ( - 1,0 ) , F _ { 2 } ( 1,0 )$ , and a line through $F _ { 2 }$ intersects $C$ at points $A , B$ . If $\left| A F _ { 2 } \right| = 2 \left| F _ { 2 } B \right|$ and $| A B | = \left| B F _ { 1 } \right|$ , then the equation of $C$ is
A. $\frac { x ^ { 2 } } { 2 } + y ^ { 2 } = 1$
B. $\frac { x ^ { 2 } } { 3 } + \frac { y ^ { 2 } } { 2 } = 1$
C. $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 1$
D. $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 4 } = 1$

12. Given that the four vertices of tetrahedron $P - A B C$ lie on the surface of sphere $O$, with $P A = P B = P C$, $\triangle A B C$ is an equilateral triangle with side length 2, $E , F$ are the midpoints of $P A , A B$ respectively, and $\angle C E F = 90 ^ { \circ }$, then the volume of sphere $O$ is
A. $8 \sqrt { 6 } \pi$
B. $4 \sqrt { 6 } \pi$
C. $2 \sqrt { 6 } \pi$
D. $\sqrt { 6 } \pi$
II. Fill-in-the-Blank Questions: This section has 4 questions, each worth 5 points, for a total of 20 points.

12. Given that the four vertices of tetrahedron $P - A B C$ lie on the surface of sphere $O$ , with $P A = P B = P C$ , $\triangle A B C$ is an equilateral triangle with side length 2, $E , F$ are the midpoints of $P A , A B$ respectively, and $\angle C E F = 90 ^ { \circ }$ , then the volume of sphere $O$ is
A. $8 \sqrt { 6 } \pi$
B. $4 \sqrt { 6 } \pi$
C. $2 \sqrt { 6 } \pi$
D. $\sqrt { 6 } \pi$
Section II: Fill-in-the-Blank Questions: This section has 4 questions, each worth 5 points, for a total of 20 points.

12. Let $F$ be the right focus of the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$, $O$ be the origin. The circle with diameter $O F$ and the circle $x ^ { 2 } + y ^ { 2 } = a ^ { 2 }$ intersect at points $P$ and $Q$. If $| P Q | = | O F |$, then the eccentricity of $C$ is
A. $\sqrt { 2 }$
B. $\sqrt { 3 }$
C. 2
D. $\sqrt { 5 }$
II. Fill-in-the-Blank Questions: This section has 4 questions, 5 points each, 20 points total.

Let $F _ { 1 } , F _ { 2 }$ be the two foci of the ellipse $C : \frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 20 } = 1$ , and $M$ be a point on $C$ in the first quadrant. If $\triangle M F _ { 1 } F _ { 2 }$ is an isosceles triangle, then the coordinates of $M$ are \_\_\_\_\_\_.

15. A quadrangular pyramid $P - A B C D$ has all vertices on the surface of sphere $O$. $PA$ is perpendicular to the plane containing rectangle $A B C D$. $AB = 3$, $AD = \sqrt { 3 }$. The surface area of sphere $O$ is $13 \pi$. The length of segment $PA$ is \_\_\_\_.

15. Let $F _ { 1 } , F _ { 2 }$ be the two foci of the ellipse $C : \frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 20 } = 1$ . Let $M$ be a point on $C$ in the first quadrant. If $\triangle M F _ { 1 } F _ { 2 }$ is an isosceles triangle, then the coordinates of $M$ are $\_\_\_\_$

15. Let $F _ { 1 } , F _ { 2 }$ be the two foci of the ellipse $C : \frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 20 } = 1$ , and let $M$ be a point on $C$ in the first quadrant. If $\triangle M F _ { 1 } F _ { 2 }$ is an isosceles triangle, then the coordinates of $M$ are $\_\_\_\_$ .

The parabola $C : x ^ { 2 } = - 2 p y$ passes through the point $( 2 , - 1 )$. (I) Find the equation of parabola $C$ and the equation of its directrix; (II) Let $O$ be the origin. A line $l$ with non-zero slope passes through the focus of parabola $C$ and intersects parabola $C$ at two points $M , N$. The line $y = - 1$ intersects lines $O M$ and $O N$ at points $A$ and $B$ respectively. Prove that the circle with $A B$ as diameter passes through two fixed points on the $y$-axis.

20. (12 points)
Let $F _ { 1 } , F _ { 2 }$ be the two foci of the ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$, $P$ be a point on $C$, and $O$ be the origin.
(1) If $\triangle P O F _ { 2 }$ is an equilateral triangle, find the eccentricity of $C$;
(2) If there exists a point $P$ such that $P F _ { 1 } \perp P F _ { 2 }$ and the area of $\triangle F _ { 1 } P F _ { 2 }$ equals 16, find the value of $b$ and the range of values for $a$.

If a circle passing through point $(2,1)$ is tangent to both coordinate axes, then the distance from the center of the circle to the line $2 x - y - 3 = 0$ is
A．$\frac { \sqrt { 5 } } { 5 }$
B．$\frac { 2 \sqrt { 5 } } { 5 }$
C．$\frac { 3 \sqrt { 5 } } { 5 }$
D．$\frac { 4 \sqrt { 5 } } { 5 }$

Let $O$ be the origin of coordinates. The line $x = 2$ intersects the parabola $C : y ^ { 2 } = 2 p x ( p > 0 )$ at points $D$ and $E$ . If $O D \perp O E$ , then the focus coordinates of $C$ are
A. $\left( \frac { 1 } { 4 } , 0 \right)$
B. $\left( \frac { 1 } { 2 } , 0 \right)$
C. $( 1,0 )$
D. $( 2,0 )$

Given the circle $x ^ { 2 } + y ^ { 2 } - 6 x = 0$ , the minimum length of the chord cut by this circle from a line passing through the point $( 1,2 )$ is
A. 1
B. 2
C. 3
D. 4

In the plane, $A , B$ are two fixed points and $C$ is a moving point. If $\overrightarrow { A C } \cdot \overrightarrow { B C } = 1$, then the locus of point $C$ is
A. a circle
B. an ellipse
C. a parabola
D. a line

If line $l$ is tangent to both the curve $y = \sqrt { x }$ and the circle $x ^ { 2 } + y ^ { 2 } = \frac { 1 } { 5 }$ , then the equation of $l$ is
A. $y = 2 x + 1$
B. $y = 2 x + \frac { 1 } { 2 }$
C. $y = \frac { 1 } { 2 } x + 1$
D. $y = \frac { 1 } { 2 } x + \frac { 1 } { 2 }$

For the ellipse $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 1$, a line $l$ passes through the right focus $F$ and intersects the ellipse at points $P$ and $Q$, with $P$ in the second quadrant. Given $Q \left( x _ { Q } , y _ { Q } \right)$ and $Q ^ { \prime } \left( x _ { Q } ^ { \prime } , y _ { Q } ^ { \prime } \right)$ both on the ellipse, with $y _ { Q } + y _ { Q } ^ { \prime } = 0$ and $F Q ^ { \prime } \perp P Q$, find the equation of line $l$ as $\_\_\_\_$

Let $A , B , C$ be three points on the surface of sphere $O$, and $\odot O _ { 1 }$ be the circumcircle of $\triangle A B C$. If the area of $\odot O _ { 1 }$ is $4 \pi$ and $A B = B C = A C = O O _ { 1 }$ , then the surface area of sphere $O$ is
A. $64 \pi$
B. $48 \pi$
C. $36 \pi$
D. $32 \pi$

Let $A , B$ be the left and right vertices of the ellipse $E : \frac { x ^ { 2 } } { a ^ { 2 } } + y ^ { 2 } = 1 ( a > 1 )$ respectively, $G$ be the upper vertex of $E$ , and $\overrightarrow { A G } \cdot \overrightarrow { G B } = 8$ . $P$ is a moving point on the line $x = 6$ , the other intersection point of $P A$ with $E$ is $C$ , and the other intersection point of $P B$ with $E$ is $D$ .
(1) Find the equation of $E$ ;
(2) Prove that the line $C D$ passes through a fixed point.

Given the ellipse $C : \frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { m ^ { 2 } } = 1 ( 0 < m < 5 )$ with eccentricity $\frac { \sqrt { 15 } } { 4 }$, where $A , B$ are the left and right vertices of $C$ respectively.
(1) Find the equation of $C$;
(2) If point $P$ is on $C$, point $Q$ is on the line $x = 6$, and $| B P | = | B Q | , B P \perp B Q$, find the area of $\triangle A P Q$ .

3. For the parabola $y ^ { 2 } = 2 p x ( p > 0 )$, the distance from its focus to the line $y = x + 1$ is $\sqrt { 2 }$. Then $p =$
A. 1
B. 2
C. $2 \sqrt { 2 }$
D. 4
【Answer】B 【Solution】 【Analysis】First determine the coordinates of the focus of the parabola, then use the point-to-line distance formula to find the value of $p$.
【Detailed Solution】The focus of the parabola has coordinates $\left( \frac { p } { 2 } , 0 \right)$. The distance from this point to the line $x - y + 1 = 0$ is: $\quad d = \frac { \left| \frac { p } { 2 } - 0 + 1 \right| } { \sqrt { 1 + 1 } } = \sqrt { 2 }$, Solving: $p = 2$ (we discard $p = -6$). Therefore, the answer is: B.

5. C

Solution: By the property of ellipses, $\left| M F _ { 1 } \right| + \left| M F _ { 2 } \right| = 2 a = 6$. By the AM-GM inequality:
$$\left| M F _ { 1 } \right| \cdot \left| M F _ { 2 } \right| \leq \frac { 1 } { 4 } \left( \left| M F _ { 1 } \right| + \left| M F _ { 2 } \right| \right) ^ { 2 } = 9 .$$
Equality holds when $\left| M F _ { 1 } \right| = \left| M F _ { 2 } \right| = 3$, and $M$ is the upper or lower vertex of the ellipse. So the answer is $C$.

11. ACD

Solution: The radius of the circle is $r = 4$. The equation of line $AB$ is $y = - \frac { 1 } { 2 } x + 2$. Drawing a line through $P$ parallel to $AB$, the distance between the two lines is $d = \frac { \left| \frac { 15 } { 2 } - 2 \right| } { \sqrt { 1 + \left( - \frac { 1 } { 2 } \right) ^ { 2 } } } = \frac { 11 } { \sqrt { 5 } }$. Since $d < 6$, the maximum distance from $P$ to line $AB$ is $d + r$, so A is correct; the minimum distance from $P$ to line $AB$ is $d - r < 2$, so B is incorrect; the extremum of $\angle P A B$ is attained when $PB$ is tangent to the circle. We have $O B = \sqrt { 5 ^ { 2 } + 3 ^ { 2 } } = \sqrt { 34 }$. Since $PB$ is tangent to $OB$, we have $PB \perp O B$. By the Pythagorean theorem, $| P B | = \sqrt { 34 - 4 ^ { 2 } } = \sqrt { 18 }$. The two tangent lines from a point to a circle have equal length, so C and D are correct. The answer is $ACD$.
When $A _ { 1 } P \perp B P$, there are two points $P$ satisfying the condition, so C is incorrect. For option D, let $E$ be the midpoint of $C C _ { 1 }$, and let $G$ be the center of rectangle $A A _ { 1 } B _ { 1 } B$. When $\mu = \frac { 1 } { 2 }$, $P$ is a point on $EF$. Since $E G \perp$ plane $A A _ { 1 } B _ { 1 }$ and $A _ { 1 } B \perp A B _ { 1 }$, we have $A _ { 1 } B \perp$ plane $E A B _ { 1 }$. When $P$ coincides with $E$, the condition is satisfied. There is a unique plane perpendicular to line $A _ { 1 } B$ passing through such points.

III. Fill in the Blank Questions

13. 1
Solution: Setting $f ( x ) = f ( - x )$ gives $x ^ { 3 } \left( 2 ^ { x } + 2 ^ { - x } \right) ( a - 1 ) = 0$ for all $x$, so $a = 1$.

15. Let $F_1, F_2$ be the two foci of the ellipse $C: \frac{x^2}{16} + \frac{y^2}{4} = 1$. Let $P, Q$ be two points on $C$ that are symmetric about the origin, and $|PQ| = |F_1F_2|$. Then the area of quadrilateral $PF_1QF_2$ is $\_\_\_\_$. [Figure]

16. Let $F _ { 1 } , F _ { 2 }$ be the two foci of the ellipse $C : \frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1$. Let $P , Q$ be two points on $C$ that are symmetric with respect to the origin, and $| P Q | = \left| F _ { 1 } F _ { 2 } \right|$. Then the area of quadrilateral $P F _ { 1 } Q F _ { 2 }$ is $\_\_\_\_$ .

III. Solution Questions: Total 70 points. Solutions should include explanations, proofs, or calculation steps. Questions 17--21 are required questions that all students must answer. Questions 22 and 23 are optional questions; students should answer according to the requirements.

(A) Required Questions: Total 60 points.

21. The parabola $C$ has its vertex at the origin $O$ and its focus on the $x$-axis. The line $l : x = 1$ intersects $C$ at points $P , Q$, and $O P \perp O Q$. Given the point $M ( 2,0 )$, and circle $\odot M$ is tangent to $l$.
(1) Find the equations of $C$ and $\odot M$;
(2) Let $A _ { 1 } , A _ { 2 } , A _ { 3 }$ be three points on $C$. Lines $A _ { 1 } A _ { 2 }$ and $A _ { 1 } A _ { 3 }$ are both tangent to $\odot M$. Determine the positional relationship between line $A _ {