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

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) $4 \sqrt { 3 }$

8. The eccentricity of hyperbola $C_1$ is $e_1$. Both the semi-major axis $a$ and semi-minor axis $b$ (where $a \neq b$) are increased by $m$ units (where $m > 0$) to obtain hyperbola $C_2$ with eccentricity $e_2$. Then
A. For any $a, b$, we have $e_1 > e_2$
B. When $a > b$, $e_1 > e_2$; when $a < b$, $e_1 < e_2$
C. For any $a, b$, we have $e_1 < e_2$
D. When $a > b$, $e_1 < e_2$; when $a < b$, $e_1 > e_2$

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 $b =$

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 $\_\_\_\_$ .

18. (This problem is worth 16 points) As shown in the figure, in the rectangular coordinate system xOy, given that the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ has eccentricity $\frac { \sqrt { 2 } } { 2 }$, and the distance from the right focus F to the left directrix l is 3. [Figure]
(1) Find the standard equation of the ellipse;
(2) A line through F intersects the ellipse at points $\mathrm { A } , \mathrm { B }$. The perpendicular bisector of segment AB intersects the line l and AB at points $\mathrm { P } , \mathrm { C }$ respectively. If $\mathrm { PC } = 2 \mathrm { AB }$, find the equation of line AB.

20. (This question is worth 13 points) The focus F of the parabola $\mathrm { C } _ { 1 } : \mathrm { X } ^ { 2 } = 4 \mathrm { y }$ is also a focus of the ellipse $\mathrm { C } _ { 2 } : \frac { y ^ { 2 } } { a ^ { 2 } } + \frac { X ^ { 2 } } { b ^ { 2 } } = 1 ( \mathrm { a } > \mathrm { b } > 0 )$. The common chord of $\mathrm { C } _ { 1 }$ and $\mathrm { C } _ { 2 }$ has length $2 \sqrt { 6 }$. A line $l$ through point F intersects $\mathrm { C } _ { 1 }$ at points $\mathrm { A } , \mathrm { B }$ and intersects $\mathrm { C } _ { 2 }$ at points $\mathrm { C } , \mathrm { D }$, with $\overrightarrow { B D }$ and $\overrightarrow { A C }$ in the same direction.
(1) Find the equation of $\mathrm { C } _ { 2 }$;
(2) If $| \mathrm { AC } | = | \mathrm { BD } |$, find the slope of line $l$.

A circle with the vertex of parabola $C$ as its center intersects $C$ at points $A , B$, and intersects the directrix of $C$ at points $D , E$. If $|DE| = 2 \sqrt { 5 }$, then the distance from the focus of $C$ to the directrix is
(A) 2
(B) 4
(C) 6
(D) 8

Given that $F$ is the right focus of the hyperbola $C: x^2 - \frac{y^2}{3} = 1$, $P$ is a point on $C$, and $PF$ is perpendicular to the $x$-axis. Point $A$ has coordinates $(1, 3)$. Then the area of $\triangle APF$ is
A. $\frac{3}{2}$
B. $\frac{1}{2}$
C. $\frac{2}{3}$
D. $\frac{3}{4}$

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. $\frac { \sqrt { 5 } } { 2 }$

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 \_\_\_\_

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 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. $y = \pm \frac { \sqrt { 3 } } { 2 } x$

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. $y = \pm \frac { \sqrt { 3 } } { 2 } x$

Given the hyperbola $C : \frac { x ^ { 2 } } { 3 } - y ^ { 2 } = 1$, with $O$ as the origin and $F$ as the right focus of $C$. A line through $F$ intersects the two asymptotes of $C$ at points $M$ and $N$. If $\triangle O M N$ is a right triangle, then $| M N | =$
A. $\frac { 3 } { 2 }$
B. 3
C. $2 \sqrt { 3 }$
D. 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. $\sqrt { 2 }$

Given point $M ( - 1, 1 )$ and parabola $C : y ^ { 2 } = 4 x$. A line through the focus of $C$ with slope $k$ intersects $C$ at points $A$ and $B$. If $\angle AMB = 90 ^ { \circ }$, then $k = $ $\_\_\_\_$.