For a hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { 6 } = 1$ with one focus at $( 3 \sqrt { 2 } , 0 )$, what is the length of the major axis? (Given that $a$ is a positive number.) [3 points]
(1) $3 \sqrt { 3 }$
(2) $\frac { 7 \sqrt { 3 } } { 2 }$
(3) $4 \sqrt { 3 }$
(4) $\frac { 9 \sqrt { 3 } } { 2 }$
(5) $5 \sqrt { 3 }$

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) $\frac { x ^ { 2 } } { 2 } - y ^ { 2 } = 1$

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 $\mathrm { MN } \perp \mathrm { AB }$.

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

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

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) $\pm \sqrt { 2 }$

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.

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 $a < b$, $e _ { 1 } < e _ { 2 }$

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. $\frac { 5 } { 3 }$

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 $\_\_\_\_$

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) $\sqrt { 2 }$

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) $4 \sqrt { 3 }$

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) $4 \sqrt { 3 }$

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

Let circle $A$ have center at $A$ and intersect one asymptote of hyperbola $C$ at points $M$ and $N$. If $\angle M A N = 60 ^ { \circ }$, then the eccentricity of $C$ is \_\_\_\_

9. If the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ has an asymptote that cuts the circle $( x - 2 ) ^ { 2 } + y ^ { 2 } = 4$ with a chord of length 2, then the eccentricity of $C$ is
A. $2$
B. $\sqrt { 3 }$
C. $\frac { 2 \sqrt { 3 } } { 3 }$
D. $\frac { \sqrt { 5 } } { 2 }$

Given an ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { 4 } = 1$ with one focus at $( 2,0 )$, then the eccentricity of $C$ is
A. $\frac { 1 } { 3 }$
B. $\frac { 1 } { 2 }$
C. $\frac { \sqrt { 2 } } { 2 }$
D. $\frac { 2 \sqrt { 2 } } { 3 }$

The hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ has eccentricity $\sqrt { 3 }$. Its asymptotes are
A. $y = \pm \sqrt { 2 } x$
B. $y = \pm \sqrt { 3 } x$
C. $y = \pm \frac { \sqrt { 2 } } { 2 } x$
D. $y = \pm \frac { \sqrt { 3 } } { 2 } x$

Let $F _ { 1 } , F _ { 2 }$ be the two foci of ellipse $C$. $P$ is a point on $C$. If $P F _ { 1 } \perp P F _ { 2 }$ and $\angle P F _ { 2 } F _ { 1 } = 60 ^ { \circ }$, then the eccentricity of $C$ is
A. $1 - \frac { \sqrt { 3 } } { 2 }$
B. $2 - \sqrt { 3 }$
C. $\frac { \sqrt { 3 } - 1 } { 2 }$
D. $\sqrt { 3 } - 1$

The eccentricity of the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ is $\sqrt { 3 }$, then its asymptote equation is
A. $y = \pm \sqrt { 2 } x$
B. $y = \pm \sqrt { 3 } x$
C. $y = \pm \frac { \sqrt { 2 } } { 2 } x$
D. $y = \pm \frac { \sqrt { 3 } } { 2 } x$

Let $F _ { 1 } , F _ { 2 }$ be the left and right foci of ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$, $A$ is the left vertex of $C$. Point $P$ is on the line passing through $A$ with slope $\frac { \sqrt { 3 } } { 6 }$. $\triangle P F _ { 1 } F _ { 2 }$ is an isosceles triangle with $\angle F _ { 1 } F _ { 2 } P = 120 ^ { \circ }$, then the eccentricity of $C$ is
A. $\frac { 2 } { 3 }$
B. $\frac { 1 } { 2 }$
C. $\frac { 1 } { 3 }$
D. $\frac { 1 } { 4 }$

Let $F _ { 1 }, F _ { 2 }$ be the left and right foci of the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ $(a > 0, b > 0)$, and $O$ be the origin. A perpendicular is drawn from $F _ { 2 }$ to an asymptote of $C$, with foot of perpendicular at $P$. If $| PF_2 | = \sqrt { 6 } | OP |$, then the eccentricity of $C$ is
A. $\sqrt { 5 }$
B. 2
C. $\sqrt { 3 }$
D. $\sqrt { 2 }$

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.

11. Let $F$ be the right focus of the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$, and $O$ be the origin. The circle with $O F$ as diameter intersects the circle $x ^ { 2 } + y ^ { 2 } = a ^ { 2 }$ at points $P , Q$. If $| P Q | = | O F |$, then the eccentricity of $C$ is
A. $\sqrt { 2 }$
B. $\sqrt { 3 }$
C. $2$
D. $\sqrt { 5 }$

Let $O$ be the origin of coordinates. The line $x = a$ intersects the two asymptotes of the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ at points $D$ and $E$ respectively. If the area of $\triangle O D E$ is 8, then the minimum value of the focal distance of $C$ is
A． 4
B． 8
C． 16
D． 32