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 $l$.

20. (This question is worth 12 points). The ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ has eccentricity $\frac { \sqrt { 2 } } { 2 }$, and the point $( 2 , \sqrt { 2 } )$ lies

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 $F$ be the focus of the parabola $C : y ^ { 2 } = 4 x$. Two perpendicular lines $l _ { 1 }$ and $l _ { 2 }$ pass through $F$. Line $l _ { 1 }$ intersects $C$ at points $A$ and $B$, and line $l _ { 2 }$ intersects $C$ at points $D$ and $E$. The minimum value of $|AB| + |DE|$ is
A. 16
B. 14
C. 12
D. 10

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 \_\_\_\_

(12 points)
Let $O$ be the origin of coordinates. Point $M$ is on the ellipse $C: \dfrac{x^2}{2} + y^2 = 1$. The perpendicular from $M$ to the $x$-axis intersects the $x$-axis at $N$. Point $P$ satisfies $\overrightarrow{NP} = \sqrt{2}\,\overrightarrow{NM}$.
(1) Find the trajectory equation of point $P$.
(2) Let point $Q$ be on the line $x = -3$, and $\overrightarrow{OP} \cdot \overrightarrow{PQ} = 1$. Prove that the line $l$ passing through point $P$ and perpendicular to $OQ$ passes through the right focus $F$ of $C$.

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

Let the parabola $C : y ^ { 2 } = 4 x$ have focus $F$. A line through $( - 2,0 )$ with slope $\frac { 2 } { 3 }$ intersects $C$ at points $M$ and $N$. Then $\overrightarrow { F M } \cdot \overrightarrow { F N } =$

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 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. $\sqrt { 3 } - 1$

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. $\sqrt { 2 }$

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. $\frac { 1 } { 4 }$

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 = $ $\_\_\_\_$.

(12 points)
Let the focus of parabola $C : y ^ { 2 } = 4 x$ be $F$. A line $l$ passing through $F$ with slope $k ( k > 0 )$ intersects $C$ at points $A$ and $B$. $| A B | = 8$.
(1) Find the equation of line $l$;
(2) Find the equation of the circle passing through points $A$ and $B$ and tangent to the directrix of $C$.

Let the parabola $C : y ^ { 2 } = 4 x$ have focus $F$. A line $l$ through $F$ with slope $k ( k > 0 )$ intersects $C$ at points $A , B$, with $| A B | = 8$.
(1) Find the equation of $l$;
(2) Find the equation of the circle passing through points $A , B$ and tangent to the directrix of $C$.

If the focus of the parabola $y ^ { 2 } = 2 p x \ ( p > 0 )$ is a focus of the ellipse $\frac { x ^ { 2 } } { 3 p } + \frac { y ^ { 2 } } { p } = 1$, then $p =$
A． 2
B． 3
C． 4
D． 8

The right focus of the hyperbola $C : \frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 2 } = 1$ is $F$. Point $P$ is on one of the asymptotes of $C$, and $O$ is the origin. If $| PO | = | PF |$, then the area of $\triangle PFO$ is
A. $\frac { 3 \sqrt { 2 } } { 4 }$
B. $\frac { 3 \sqrt { 2 } } { 2 }$
C. $2 \sqrt { 2 }$
D. $3 \sqrt { 2 }$

10. Let $F$ be a focus of the hyperbola $C : \frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 5 } = 1$ . Point $P$ is on $C$ , $O$ is the origin. If $| O P | = | O F |$ , then the area of $\triangle O P F$ is
A. $\frac { 3 } { 2 }$
B. $\frac { 5 } { 2 }$
C. $\frac { 7 } { 2 }$
D. $\frac { 9 } { 2 }$

10. For the hyperbola $C : \frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 2 } = 1$ with right focus $F$ , if point $P$ is on one of the asymptotes of $C$ , $O$ is the origin, and $| P O | = | P F |$ , then the area of $\triangle P F O$ is
A. $\frac { 3 \sqrt { 2 } } { 4 }$
B. $\frac { 3 \sqrt { 2 } } { 2 }$
C. $2 \sqrt { 2 }$
D. $3 \sqrt { 2 }$

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. $\sqrt { 5 }$

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 \_\_\_\_\_\_.