18. The ellipse $E : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ ($a > b > 0$) passes through the point $(0, \sqrt { 2 })$, and has eccentricity [Figure]
(1) Find the equation of ellipse $E$;
(2) The line $x = m y - 1$ ($m \in \mathbb{R}$) intersects the ellipse $E$ at points $A$ and $B$. Determine the positional relationship between the point $G \left( - \frac { 9 } { 4 } , 0 \right)$ and the circle with diameter $AB$, and explain the reason.

(1) Find the equation of ellipse $E$;
(1) Find the equation of ellipse $E$;

(2) The line $x = m y - 1$ ($m \in \mathbb{R}$) intersects the ellipse $E$ at points $A$ and $B$. Determine the positional relationship between the point $G \left( - \frac { 9 } { 4 } , 0 \right)$ and the circle with diameter $AB$, and explain the reason. [Figure]

19. (This question is worth 14 points) Given the ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ with eccentricity $\frac { \sqrt { 2 } } { 2 }$, point $P ( 0,1 )$ and point $A ( m , n ) ( m \neq 0 )$ are both on the ellipse $C$. The line $P A$ intersects the $x$-axis at point $M$. (I) Find the equation of ellipse $C$ and find the coordinates of point $M$ (expressed in terms of $m , n$); (II) Let $O$ be the origin. Point $B$ is symmetric to point $A$ with respect to the $x$-axis. The line $P B$ intersects the $x$-axis at point $N$. Question: Does there exist a point $Q$ on the $y$-axis such that $\angle O Q M = \angle O N Q$? If it exists, find the coordinates of point $Q$; if it does not exist, explain the reason.

19. Given the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ with upper vertex $B$, left focus $F$, and eccentricity $\frac { \sqrt { 5 } } { 5 }$.
(1) Find the slope of line $BF$;
(2) Let line $BF$ intersect the ellipse at point $P$ (where $P$ is different from $B$). A line passing through $B$ and perpendicular to $BF$ intersects the ellipse at point $Q$ (where $Q$ is different from $B$). Line $PQ$ intersects the $x$-axis at point $M$, and $|PM| = l|MQ|$.
1) Find the value of $l$;
2) If $|PM| \sin \angle BQP = \frac { 7 \sqrt { 5 } } { 9 }$, find the equation of the ellipse.

Given an ellipse with left focus $\mathrm{F}(-c, 0)$ and eccentricity $\frac{\sqrt{3}}{3}$. Point M is on the ellipse and in the first quadrant. The line segment of line FM intercepted by the circle $x^2 + y^2 = \frac{b^2}{4}$ has length c, and $|FM| = \frac{4\sqrt{3}}{3}$.
(I) Find the slope of line FM;
(II) Find the equation of the ellipse;
(III) Let P be a moving point on the ellipse. If the slope of line FP is greater than $\sqrt{2}$, find the range of the slope of line OP

19. (15 points) As shown in the figure, given the parabola $\mathrm { C } _ { 1 } : \mathrm { y } = \frac { 1 } { 4 } x ^ { 2 }$ , the circle $\mathrm { C } _ { 2 } : x ^ { 2 } + ( \mathrm { y } - 1 ) ^ { 2 } = 1$ , through point $\mathrm { P } ( \mathrm { t } , 0 ) ( \mathrm { t } > 0 )$ , draw lines $\mathrm { PA } , \mathrm { PB}$ not passing through the origin O that are tangent to the parabola $C _ { 1 }$ and circle $\mathrm { C } _ { 2 }$ respectively, with $\mathrm { A } , \mathrm { B}$ as the points of tangency.
(1) Find the coordinates of points $\mathrm { A } , \mathrm { B}$ ;
(2) Find the area of $\triangle \mathrm { PAB}$ . Note: If a line has exactly one common point with a parabola and is not parallel to the axis of symmetry of the parabola, then the line is tangent to the parabola, and the common point is called the point of tangency. [Figure]

19. (This question is worth 15 points) Two distinct points $A , B$ on the ellipse $\frac { x ^ { 2 } } { 2 } + y ^ { 2 } = 1$ are symmetric about the line $y = mx + \frac { 1 } { 2 }$ . (I) Find the range of the real number $m$; (II) Find the maximum value of the area of $\triangle AOB$ (where $O$ is the origin). [Figure]

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 the ellipse $C : x ^ { 2 } + 3 y ^ { 2 } = 3$, a line passing through point $D ( 1,0 )$ but not through point $E ( 2,1 )$ intersects the ellipse $C$ at points $A$ and $B$. The line $AE$ intersects the line $x = 3$ at point $M$.\n(1) Find the eccentricity of the ellipse $C$;\n(II) If $AB$ is perpendicular to the $x$-axis, find the slope of line $BM$;\n(III) Determine the positional relationship between line $BM$ and line $DE$, and explain the reason.

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

20. (12 points) The ellipse $\mathrm { E } : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ has semi-focal distance $c$. The distance from the origin O to the line passing through the points $( c , 0 )$ and $( 0 , b )$ is $\frac { 1 } { 2 } c$. (I) Find the eccentricity of ellipse E; (II) As shown in the figure, AB is a diameter of circle $\mathrm { M } : ( x + 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = \frac { 5 } { 2 }$. If the ellipse E passes through points A and B, find the equation of ellipse E. [Figure]

20. (This question is worth 13 points) As shown in the figure, the ellipse $\mathrm { E } : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ has eccentricity $\frac { \sqrt { 2 } } { 2 }$. The point $( 0,1 )$ is on the minor axis $CD$, and $\overline { P C } \overline { P D } = - 1$. (I) Find the equation of ellipse E; (II) Let O be the origin. A moving line through point P intersects the ellipse at points A and B. Does there exist a constant $\lambda$ such that $\overline { O A } \overline { O B } + \lambda \overline { P A } \overline { P B }$ is a constant? If it exists, find the value of $\lambda$; if it does not exist, explain why. [Figure]

20. As shown in the figure, the ellipse $\mathrm { E } : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ $(a > b > 0)$ has eccentricity $\frac { \sqrt { 2 } } { 2 }$. A moving line $l$ passing through point $\mathrm { P } ( 0, 1 )$ intersects the ellipse at points $\mathrm { A }$ and $\mathrm { B }$. When line $l$ is parallel to the $x$-axis, the length of the chord intercepted by line $l$ on ellipse $E$ is $2 \sqrt { 2 }$.
(1) Find the equation of ellipse $E$;
(2) In the rectangular coordinate system $xOy$, does there exist a fixed point $Q$ different from point $P$ such that $\frac { | Q A | } { | Q B | } = \frac { | P A | } { | P B | }$ always holds? If it exists, find the coordinates of point $Q$; if it does not exist, explain the reason. [Figure]

21. (Multiple Choice) This problem includes four sub-problems A, B, C, and D. Please select two of them and answer in the corresponding areas. If more are done, the first two sub-problems will be graded. When solving, you should write out text explanations, proofs, or calculation steps.
A. [Elective 4-1: Geometric Proof Selection] (This problem is worth 10 points) In $\triangle A B C$, $A B = A C$, the chord $A E$ of the circumcircle O of $\triangle A B C$ intersects $B C$ at point D. Prove: $\triangle A B D \sim \triangle A E B$ [Figure]
B. [Elective 4-2: Matrices and Transformations] (This problem is worth 10 points) Given $x , y \in R$, the vector $\alpha = \left[ \begin{array} { c } 1 \\ - 1 \end{array} \right]$ is an eigenvector of the matrix $A = \left[ \begin{array} { c c } x & 1 \\ y & 0 \end{array} \right]$ corresponding to the eigenvalue $- 2$. Find the matrix A and its other eigenvalue.

C. [Elective 4-4: Coordinate Systems and Parametric Equations]

The polar equation of circle C is $\rho ^ { 2 } + 2 \sqrt { 2 } \rho \sin \left( \theta - \frac { \pi } { 4 } \right) - 4 = 0$. Find the radius of circle C.
D. [Elective 4-5: Inequalities Selection] Solve the inequality $x + | 2 x + 3 | \geq 3$

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

(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 $C$.

The line $x + y + 2 = 0$ intersects the $x$-axis and $y$-axis at points $A$ and $B$ respectively. Point $P$ is on the circle $( x - 2 ) ^ { 2 } + y ^ { 2 } = 2$. The range of the area of $\triangle ABP$ is
A. $[ 2,6 ]$
B. $[ 4,8 ]$
C. $[ \sqrt { 2 } , 3 \sqrt { 2 } ]$
D. $[ 2 \sqrt { 2 } , 3 \sqrt { 2 } ]$

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 $\overrightarrow { F M } \cdot \overrightarrow { F N } =$

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 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. $\frac { 1 } { 4 }$

The line $y = x + 1$ intersects the circle $x ^ { 2 } + y ^ { 2 } + 2 y - 3 = 0$ at points $A$ and $B$. Then $| A B | = $ \_\_\_\_

(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 $C$.

Given that the eccentricity of the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ is $\frac { 1 } { 2 }$, then (A) $a ^ { 2 } = 2 b ^ { 2 }$ (B) $3 a ^ { 2 } = 4 b ^ { 2 }$ (C) $a = 2 b$ (D) $3 a = 4 b$

There are many beautifully shaped and meaningful curves in mathematics. The curve $C : x ^ { 2 } + y ^ { 2 } = 1 + | x | y$ is one of them (as shown in the figure). Three conclusions are given: (1) The curve $C$ passes through exactly 6 lattice points (points with both integer coordinates); (2) The distance from any point on curve $C$ to the origin does not exceed $\sqrt { 2 }$; (3) The area of the ``heart-shaped'' region enclosed by curve $C$ is less than 3. The sequence numbers of all correct conclusions are (A) (1) (B) (2) (C) (1)(2) (D) (1)(2)(3)

9. 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