For two positive numbers $a , p$, let $\mathrm { F } _ { 1 }$ be the focus of the parabola $( y - a ) ^ { 2 } = 4 p x$, and let $\mathrm { F } _ { 2 }$ be the focus of the parabola $y ^ { 2 } = - 4 x$.
When segment $\mathrm { F } _ { 1 } \mathrm {~F} _ { 2 }$ meets the two parabolas at points $\mathrm { P } , \mathrm { Q }$ respectively, $\overline { \mathrm { F } _ { 1 } \mathrm {~F} _ { 2 } } = 3$ and $\overline { \mathrm { PQ } } = 1$. What is the value of $a ^ { 2 } + p ^ { 2 }$? [4 points]
(1) 6
(2) $\frac { 25 } { 4 }$
(3) $\frac { 13 } { 2 }$
(4) $\frac { 27 } { 4 }$
(5) 7

Let F be the focus of the parabola $y^2 = 8x$. From a point A on the parabola, drop a perpendicular to the directrix of the parabola, with the foot of the perpendicular being B. Let C and D be the two points where the line BF intersects the parabola. When $\overline{\mathrm{BC}} = \overline{\mathrm{CD}}$, what is the area of triangle ABD? (where $\overline{\mathrm{CF}} < \overline{\mathrm{DF}}$ and point A is not the origin) [3 points]
(1) $100\sqrt{2}$
(2) $104\sqrt{2}$
(3) $108\sqrt{2}$
(4) $112\sqrt{2}$
(5) $116\sqrt{2}$

As shown in the figure, there are two distinct planes $\alpha$ and $\beta$ with intersection line containing two points $\mathrm{A}$ and $\mathrm{B}$ where $\overline{\mathrm{AB}} = 18$. A circle $C_1$ with diameter AB lies on plane $\alpha$, and an ellipse $C_2$ with major axis AB and foci $\mathrm{F}$ and $\mathrm{F'}$ lies on plane $\beta$. Let H be the foot of the perpendicular from a point P on circle $C_1$ to plane $\beta$. Given that $\overline{\mathrm{HF'}} < \overline{\mathrm{HF}}$ and $\angle\mathrm{HFF'} = \frac{\pi}{6}$. Let Q be the point on ellipse $C_2$ where line HF intersects it, closer to H, with $\overline{\mathrm{FH}} < \overline{\mathrm{FQ}}$. The circle on plane $\beta$ centered at H passing through Q has radius 4 and is tangent to line AB. If the angle between the two planes $\alpha$ and $\beta$ is $\theta$, what is the value of $\cos\theta$? (where point P is not on plane $\beta$) [4 points]
(1) $\frac{2\sqrt{66}}{33}$
(2) $\frac{4\sqrt{69}}{69}$
(3) $\frac{\sqrt{2}}{3}$
(4) $\frac{4\sqrt{3}}{15}$
(5) $\frac{2\sqrt{78}}{39}$

For a positive number $c$, there is a hyperbola with foci $\mathrm{F}(c, 0)$ and $\mathrm{F'}(-c, 0)$ and major axis length 6. Two distinct points $\mathrm{P}$ and $\mathrm{Q}$ on this hyperbola satisfy the following conditions. Find the sum of all values of $c$. [4 points] (가) Point P is in the first quadrant, and point Q is on line $\mathrm{PF'}$. (나) Triangle $\mathrm{PF'F}$ is isosceles. (다) The perimeter of triangle PQF is 28.

For a natural number $n$ ($n \geq 2$), let the line $x = \frac{1}{n}$ meet the two ellipses $$C_{1} : \frac{x^{2}}{2} + y^{2} = 1, \quad C_{2} : 2x^{2} + \frac{y^{2}}{2} = 1$$ at points P and Q respectively in the first quadrant. Let $\alpha$ be the $x$-intercept of the tangent line to ellipse $C_{1}$ at point P, and let $\beta$ be the $x$-intercept of the tangent line to ellipse $C_{2}$ at point Q. How many values of $n$ satisfy $6 \leq \alpha - \beta \leq 15$? [3 points]
(1) 7
(2) 9
(3) 11
(4) 13
(5) 15

There is a hyperbola $x^{2} - \frac{y^{2}}{35} = 1$ with foci at $\mathrm{F}(c, 0)$, $\mathrm{F}'(-c, 0)$ ($c > 0$). For a point P on this hyperbola in the first quadrant, let Q be a point on line $\mathrm{PF}'$ such that $\overline{\mathrm{PQ}} = \overline{\mathrm{PF}}$. When triangle $\mathrm{QF'F}$ and triangle $\mathrm{FF'P}$ are similar, the area of triangle PFQ is $\frac{q}{p}\sqrt{5}$. Find the value of $p + q$. (Here, $\overline{\mathrm{PF}'} < \overline{\mathrm{QF}'}$ and $p$ and $q$ are coprime natural numbers.) [4 points]

3. If the hyperbola $E : \frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$ has left and right foci $F _ { 1 }$ and $F _ { 2 }$ respectively, point $P$ is on the hyperbola $E$, and $\left| P F _ { 1 } \right| = 3$, then $\left| P F _ { 2 } \right|$ equals
A. 11
B. 9
C. 5
D. 3

5. A line passing through the right focus of the hyperbola $x ^ { 2 } - \frac { y ^ { 2 } } { 3 } = 1$ and perpendicular to the $x$-axis intersects the two asymptotes of the hyperbola at points $\mathrm { A }$ and $\mathrm { B }$, then $| A B | =$
(A) $\frac { 4 \sqrt { 3 } } { 3 }$
(B) $2 \sqrt { 3 }$
(C) $6$
(D) $4 \sqrt { 3 }$

5. Given the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ with one focus at $F ( 2,0 )$, and the asymptote of the hyperbola is tangent to the circle $( x - 2 ) ^ { 2 } + y ^ { 2 } = 3$, then the equation of the hyperbola is
(A) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 13 } = 1$
(B) $\frac { x ^ { 2 } } { 13 } - \frac { y ^ { 2 } } { 9 } = 1$
(C) $\frac { x ^ { 2 } } { 3 } - y ^ { 2 } = 1$

6. Among the following hyperbolas, which one has asymptote equations $y = \pm 2 x$?
(A) $x ^ { 2 } - \frac { y ^ { 2 } } { 4 } = 1$
(B) $\frac { x ^ { 2 } } { 4 } - y ^ { 2 } = 1$
(C) $x ^ { 2 } - \frac { y ^ { 2 } } { 2 } = 1$
(D) $\frac { x ^ { 2 } } { 2 } - y ^ { 2 } = 1$

6. If one asymptote of the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ passes through the point $( 3 , - 4 )$, then the eccentricity of this hyperbola is
A. $\frac { \sqrt { 7 } } { 3 }$
B. $\frac { 5 } { 4 }$
C. $\frac { 4 } { 3 }$
D. $\frac { 5 } { 3 }$

Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ $(a > 0, b > 0)$, one of its asymptotes passes through the point $(2, \sqrt{3})$, and one focus of the hyperbola lies on the directrix of the parabola $y^2 = 4\sqrt{7}x$. Then the equation of the hyperbola is
(A) $\frac{x^2}{21} - \frac{y^2}{28} = 1$
(B) $\frac{x^2}{28} - \frac{y^2}{21} = 1$
(C) $\frac{x^2}{3} - \frac{y^2}{4} = 1$
(D) $\frac{x^2}{4} - \frac{y^2}{3} = 1$

7. A line passing through the right focus of the hyperbola $x ^ { 2 } - \frac { y ^ { 2 } } { 3 } = 1$ and perpendicular to the $x$-axis intersects the two asymptotes of the hyperbola at points $A$ and $B$. Then $| A B | =$ [Figure]
(A) $\frac { 4 \sqrt { 3 } } { 3 }$
(B) $2 \sqrt { 3 }$
(C) 6
(D) $4 \sqrt { 3 }$

For the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( \mathrm { a } > 0 , \mathrm {~b} > 0 )$, let $F$ be the right focus, and $\mathrm { A } _ { 1 } , \mathrm {~A} _ { 2 }$ be the left and right vertices respectively. A line through $F$ perpendicular to $A _ { 1 } A _ { 2 }$ intersects the hyperbola at points $B$ and $C$. If $A _ { 1 } B \perp A _ { 2 } C$, then the slope of the asymptotes of the hyperbola is
(A) $\pm \frac { 1 } { 2 }$
(B) $\pm \frac { \sqrt { 2 } } { 2 }$
(C) $\pm 1$
(D) $\pm \sqrt { 2 }$

9. The real semi-major axis length a and imaginary semi-minor axis length $\mathrm { b }$ of a hyperbola $C _ { 1 }$ with eccentricity $e _ { 1 }$ (where $a = b$) are both increased by $\mathrm { m } ( m > 0 )$ units of length to obtain a hyperbola $C _ { 2 }$ with eccentricity $e _ { 2 }$. Then
A. For any $\mathrm { a } , \mathrm { b }$, $e _ { 1 } < e _ { 2 }$
B. When $a > b$, $e _ { 1 } < e _ { 2 }$ ; when $a < b$, $e _ { 1 } > e _ { 2 }$
C. For any a, b, $e _ { 1 } > e _ { 2 }$
D. When $a > b$, $e _ { 1 } > e _ { 2 }$ ; when $a < b$, $e _ { 1 } < e _ { 2 }$

9. The focal distance of the hyperbola $\frac { x ^ { 2 } } { 2 } - y ^ { 2 } = 1$ is $\_\_\_\_$ , and the equations of the asymptotes are $\_\_\_\_$ .

10. Given that the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - y ^ { 2 } = 1 ( a > 0 )$ has an asymptote $\sqrt { 3 } x + y = 0$, then $a =$ $\_\_\_\_$.

10. Let the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( \mathrm { a } > 0 , \mathrm {~b} > 0 )$ have right focus $F$. A line through $F$ perpendicular to $AF$ intersects the hyperbola at points $\mathrm { B }$ and $\mathrm { C }$. Lines through $\mathrm { B }$ and $\mathrm { C }$ perpendicular to $\mathrm { AC }$ and $\mathrm { AB }$ respectively intersect at point $D$. If the distance from $D$ to line $BC$ is less than $a + \sqrt { a ^ { 2 } + b ^ { 2 } }$, then the range of the slope of the asymptotes of the hyperbola is
A. $( - 1,0 ) \cup ( 0,1 )$
B. $( - \infty , - 1 ) \cup \left( 1 , + \infty \right)$
C. $( - \sqrt { 2 } , 0 ) \cup ( 0 , \sqrt { 2 } )$
D. $( - \infty , - \sqrt { 2 } ) \cup ( \sqrt { 2 } , + \infty )$
II. Fill-in-the-Blank Questions: This section contains 6 questions. Candidates answer 5 of them, each worth 5 points, for a total of 25 points. Write your answers in the corresponding positions on the answer sheet.

Points $A$ and $B$ are the left and right vertices of hyperbola $E$. Point $M$ is on $E$, and $\triangle A B M$ is an isosceles triangle with vertex angle $120 ^ { \circ }$. Then the eccentricity of $E$ is
(A) $\sqrt { 5 }$
(B) $2$
(C) $\sqrt { 3 }$
(D) $\sqrt { 2 }$

Given that $(2,0)$ is a focus of the hyperbola $x ^ { 2 } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ $(b > 0)$, then $b =$

12. In the rectangular coordinate system $x O y$, let $P$ be a moving point on the right branch of the hyperbola $x ^ { 2 } - y ^ { 2 } = 1$. If the distance from point $P$ to the line $x - y + 1 = 0$ is always greater than or equal to c, then the maximum value of the real number c is $\_\_\_\_$.

14. Let F be a focus of the hyperbola $C: \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$. If there exists a point P on C such that the midpoint of segment PF is exactly an endpoint of its conjugate axis, then the eccentricity of C is $\_\_\_\_$

14. If the directrix of the parabola $y ^ { 2 } = 2 p x ( p > 0 )$ passes through a focus of the hyperbola $x ^ { 2 } - y ^ { 2 } = 1$, then $p = $ $\_\_\_\_$

15. A hyperbola passes through the point $( 4 , \sqrt { 3 } )$ and has asymptote equations $y = \pm \frac { 1 } { 2 } x$. The standard equation of this hyperbola is $\_\_\_\_$ .

20. Let the equation of ellipse E be $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$. Let O be the origin, point A has coordinates $( a , 0 )$, point B has coordinates $( 0 , b )$. Point M is on the line segment AB and satisfies $| B M | = 2 | M A |$. The slope of line OM is $\frac { \sqrt { 5 } } { 10 }$.
(1) Find the eccentricity $e$ of E;
(2) Let point C have coordinates $( 0 , - \mathrm { b } )$, and N be the midpoint of segment AC. Prove that $\mathrm { MN } \perp \mathrm { AB }$.