Let $O$ be the origin of coordinates. The line $y=-\sqrt{3}(x-1)$ passes through the focus of the parabola $C: y^2=2px$ $(p>0)$ and intersects $C$ at points $M$ and $N$. Let $l$ be the directrix of $C$. Then
A. $p=2$
B. $|MN|=\frac{8}{3}$
C. the circle with $MN$ as diameter is tangent to $l$
D. $\triangle OMN$ is an isosceles triangle

Let $A$ and $B$ be two points on the hyperbola $x ^ { 2 } - \frac { y ^ { 2 } } { 9 } = 1$. Which of the following four points could be the midpoint of segment $AB$?
A. $(1,1)$
B. $( - 1,2 )$
C. $( 1,3 )$
D. $( - 1 , - 4 )$

The ellipse $\frac{x^{2}}{9} + \frac{y^{2}}{6} = 1$ has foci $F_{1} , F_{2}$ and center $O$ . Let $P$ be a point on the ellipse. If $\cos \angle F_{1}PF_{2} = \frac{3}{5}$ , then $|OP| =$
A. $\frac{2}{5}$
B. $\frac{\sqrt{30}}{2}$
C. $\frac{3}{5}$
D. $\frac{\sqrt{35}}{2}$

Given that point $A ( 1 , \sqrt { 5 } )$ lies on the parabola $C : y ^ { 2 } = 2 p x$, then the distance from $A$ to the directrix of $C$ is \_\_\_\_

The line $x - 2y + 1 = 0$ intersects the parabola $y^{2} = 2px \ (p > 0)$ at points $A , B$ with $AB = 4\sqrt{15}$ .
(1) Find the value of $p$ ;
(2) Let $F$ be the focus of $y^{2} = 2px$ . Let $M , N$ be two points on the parabola such that $\overrightarrow{MF} \perp \overrightarrow{NF}$ . Find the minimum area of $\triangle MNF$ .

Given the ellipse $C : \frac { y ^ { 2 } } { a ^ { 2 } } + \frac { x ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ with eccentricity $\frac { \sqrt { 5 } } { 3 }$, and point $A ( - 2,0 )$ lies on $C$.
(1) Find the equation of $C$.
(2) A line passing through point $( - 2,3 )$ intersects $C$ at points $P$ and $Q$. Lines $AP$ and $AQ$ intersect the $y$-axis at points $M$ and $N$ respectively. Prove that the midpoint of segment $MN$ is a fixed point.

The directrix of parabola $C : y ^ { 2 } = 4 x$ is $l$. Let $P$ be a moving point on $C$. Draw a tangent line to circle $\odot A : x ^ { 2 } + ( y - 4 ) ^ { 2 } = 1$ through $P$, with $Q$ as the point of tangency. Draw a perpendicular from $P$ to line $l$, with $B$ as the foot of the perpendicular. Then
A. Line $l$ is tangent to $\odot A$
B. When $P , A , B$ are collinear, $| P Q | = \sqrt { 15 }$
C. When $| P B | = 2$, $P A \perp A B$
D. There are exactly 2 points $P$ satisfying $| P A | = | P B |$

Given the parabola $y ^ { 2 } = 16 x$, the coordinates of the focus are \_\_\_\_.

The shape ``$\varnothing$'' can be made into a beautiful ribbon. Consider it as part of the curve $C$ in the figure. It is known that $C$ passes through the origin $O$ , and points on $C$ satisfy: the abscissa is greater than $- 2$ , and the product of the distance to point $F ( 2,0 )$ and the distance to the line $x = a ( a < 0 )$ equals 4 . Then
A. $a = - 2$
B. The point $( 2 \sqrt { 2 } , 0 )$ is on $C$
C. The maximum ordinate of points on $C$ in the first quadrant is 1
D. When point $\left( x _ { 0 } , y _ { 0 } \right)$ is on $C$ , $y _ { 0 } \leqslant \frac { 4 } { x _ { 0 } + 2 }$

Let the hyperbola $C : \frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ have left and right foci $F _ { 1 } , F _ { 2 }$ respectively. A line through $F _ { 2 }$ parallel to the $y$-axis intersects $C$ at points $A$ and $B$ . If $\left| F _ { 1 } A \right| = 13 , | A B | = 10$ , then the eccentricity of $C$ is $\_\_\_\_$ .

Given the hyperbola $\frac { x ^ { 2 } } { 4 } - y ^ { 2 } = 1$, find the slopes of lines passing through $( 3,0 )$ that have only one intersection point with the hyperbola \_\_\_\_.

(15 points) Given that $A ( 0,3 )$ and $P \left( 3 , \frac { 3 } { 2 } \right)$ are two points on the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ .
(1) Find the eccentricity of $C$ ;
(2) If a line $l$ through $P$ intersects $C$ at another point $B$ , and the area of $\triangle A B P$ is 9 , find the equation of $l$ .

Given the ellipse equation $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$. The foci and endpoints of the minor axis form a square with side length 2. A line $l$ passing through $( 0 , t ) ( t > \sqrt { 2 })$ intersects the ellipse at points $A, B$, and $C ( 0,1 )$. Connect $AC$ and it intersects the ellipse at $D$.
(1) Find the equation of the ellipse and its eccentricity;
(2) If the slope of line $BD$ is 0, find $t$.

Given hyperbola $C : x ^ { 2 } - y ^ { 2 } = m ( m > 0 )$, point $P _ { 1 } ( 5,4 )$ is on $C$, and $k$ is a constant with $0 < k < 1$. Through $P_n$ on the right branch of $C$, draw a line with slope $k$; this line intersects $C$ at another point $Q_n$ on the left branch. The reflection of $Q_n$ across the $y$-axis gives $P_{n+1}$ on the right branch.
(1) Find the coordinates of $P_2$ when $k = \frac{1}{2}$.
(2) Prove that $\{x_n - y_n\}$ is a geometric sequence.
(3) Prove that $S_n = \sum_{i=1}^{n} (x_i y_{i+1} - y_i x_{i+1})$ is a constant independent of $n$.

If the imaginary axis length of hyperbola $C$ is $\sqrt{7}$ times the real axis length, then the eccentricity of $C$ is
A. $\sqrt{2}$
B. $2$
C. $\sqrt{7}$
D. $2\sqrt{2}$

If the imaginary axis length of hyperbola $C$ is $\sqrt{7}$ times the real axis length, then the eccentricity of $C$ is
A. $\sqrt{2}$
B. $2$
C. $\sqrt{7}$
D. $2\sqrt{2}$

Let the parabola $C: y^2 = 2px$ $(p > 0)$ have focus $F$. Point $A$ is on $C$. A perpendicular is drawn from $A$ to the directrix of $C$, with foot $B$. If the equation of line $BF$ is $y = -2x + 2$, then $|AF| = $ ( )
A. $3$
B. $4$
C. $5$
D. $6$

Let the focus of parabola $C: y^2 = 6x$ be $F$. A line through $F$ intersects $C$ at $A$ and $B$. A line through $F$ perpendicular to $AB$ intersects the directrix $l: x = -\frac{3}{2}$ at $E$. From point $A$, draw a perpendicular to the directrix $l$ with foot $D$. Then
A. $|AD| = |AF|$
B. $|AE| = |AB|$
C. $|AB| \geq 6$
D. $|AE| \cdot |BE| \geq 18$

Let the focus of parabola $C: y^2 = 6x$ be $F$. A line through $F$ intersects $C$ at $A$ and $B$. A perpendicular from $A$ to the line $l: x = -\frac{3}{2}$ meets it at $D$. A line through $F$ perpendicular to $AB$ meets $l$ at $E$. Then
A. $|AD| = |AF|$
B. $|AE| = |AB|$
C. $|AB| \geq 6$
D. $|AE| \cdot |BE| \geq 18$

For the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ $(a > 0, b > 0)$, let $F_1, F_2$ be the left and right foci respectively, and $A_1, A_2$ be the left and right vertices respectively. The circle with diameter $F_1F_2$ intersects one asymptote of $C$ at points $M, N$, and $\angle NA_1M = \frac{5\pi}{6}$, then
A. $\angle A_1MA_2 = \frac{\pi}{6}$
B. $|MA_1| = 2|MA_2|$
C. The eccentricity of $C$ is $\sqrt{13}$
D. When $a = \sqrt{2}$, the area of quadrilateral $NA_1MA_2$ is $8\sqrt{3}$

Let the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ have eccentricity $\frac{2\sqrt{2}}{3}$, with lower vertex $A$ and right vertex $B$, $|AB| = \sqrt{10}$.
(1) Find the standard equation of the ellipse.
(2) Let $P$ be a moving point not on the $y$-axis, and let $R$ be a point on the ray $AP$ satisfying $|AR| \cdot |AP| = 3$.
(i) If $P(m, n)$, find the coordinates of point $R$ (expressed in terms of $m, n$).
(ii) Let $O$ be the origin, and $Q$ be a moving point on $C$. The slope of line $OR$ is 3 times the slope of line $OP$. Find the maximum value of $|PQ|$.

(17 points) Given the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ with eccentricity $\frac{2\sqrt{2}}{3}$, lower vertex $A$, right vertex $B$, and $|AB| = \sqrt{10}$.
(1) Find the equation of $C$.
(2) Let $P$ be a moving point not on the $y$-axis, and let $R$ be a point on the ray $AP$ satisfying $|AP| \cdot |AR| = 3$.
(i) If $P(m, n)$, find the coordinates of $R$ (expressed in terms of $m, n$).
(ii) Let $O$ be the origin, and $Q$ be a moving point on $C$. The slope of line $OR$ is 3 times the slope of line $OP$. Find the maximum value of $|PQ|$.

Throughout this part $A$ and $B$ denote real symmetric matrices of $\mathcal { M } _ { 2 } ( \mathbb { R } )$. We denote by $\lambda _ { 1 } \leqslant \lambda _ { 2 }$ the eigenvalues of $A$.
We consider the set $\Gamma \subset \mathbb { R } ^ { 2 }$ defined by the equation $\langle A X , X \rangle = 1$.
II.B.1) Characterize the conditions on the $\lambda _ { i }$ for which this set is: a) empty; b) the union of two lines; c) an ellipse; d) a hyperbola.
II.B.2) Represent on the same figure the sets $\Gamma$ obtained for $A$ diagonal with $\lambda _ { 1 } \in \{ - 4 , - 1,0,1 / 4,1 \}$ and $\lambda _ { 2 } = 1$.

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$, let $\mathcal{H}_A$ be the quadric with equation $\psi_A(x,y,z) = 0$ where $\psi_A(x,y,z)$ is the real part of the determinant of $\left(\begin{array}{cc} a-x-\mathrm{i}z & b-y \\ c+y & d-x-\mathrm{i}z \end{array}\right)$.
Specify the intersection $Z_A$ of $\mathcal{H}_A$ with the plane with equation $x = (a+d)/2$.

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$, let $\mathcal{H}_A$ be the quadric with equation $\psi_A(x,y,z) = 0$ where $\psi_A(x,y,z)$ is the real part of the determinant of $\left(\begin{array}{cc} a-x-\mathrm{i}z & b-y \\ c+y & d-x-\mathrm{i}z \end{array}\right)$, and let $Z_A$ be the intersection of $\mathcal{H}_A$ with the plane $x = (a+d)/2$.
If the matrix $A$ has two non-real eigenvalues, how can one see the eigenvalues of $A$ on $\mathcal{H}_A$? (One may consider the intersection of $Z_A$ with the plane with equation $y = 0$.) Can one see a basis of eigenvectors using $\mathcal{H}_A$?