The equation of a circle with center $(1,1)$ and passing through the origin is

As shown in the figure, in circle O, M and N are trisection points of chord AB. Chords CD and CE pass through points M and N respectively. If $\mathrm{CM} = 2$, $\mathrm{MD} = 4$, $\mathrm{CN} = 3$, then the length of segment NE is
(A) $\frac{8}{3}$
(B) 3
(C) $\frac{10}{3}$
(D) $\frac{5}{2}$

5. As shown in the figure, let $F$ be the focus of the parabola $y ^ { 2 } = 4 x$. A line not passing through the focus contains three distinct points $A , B , C$, where points $A , B$ are on the parabola and point $C$ is on the $y$-axis. Then the ratio of the areas of $\triangle BCF$ and $\triangle ACF$ is
A. $\frac { | B F | - 1 } { | A F | - 1 }$
B. $\frac { | B F | ^ { 2 } - 1 } { | A F | ^ { 2 } - 1 }$
C. $\frac { | B F | + 1 } { | A F | + 1 }$
D. $\frac { | B F | ^ { 2 } + 1 } { | A F | ^ { 2 } + 1 }$

7. Given three points $A ( 1,0 ) , B ( 0 , \sqrt { 3 } ) , C ( 2 , \sqrt { 3 } )$, the distance from the circumcenter of $\triangle A B C$ [Figure]
to the origin is
A. $\frac { 5 } { 3 }$
B. $\frac { \sqrt { 21 } } { 3 }$
C. $\frac { 2 \sqrt { 5 } } { 3 }$
D. $\frac { 4 } { 3 }$

The circle passing through three points $A ( 1,3 ) , B ( 4,2 ) , C ( 1,7 )$ intersects the $y$-axis at points $\mathrm { M }$ and $\mathrm { N }$. Then $| M N | =$
(A) $2 \sqrt { 6 }$
(B) $8$
(C) $4 \sqrt { 6 }$
(D) $10$

8. The line $3 \mathrm { x } + 4 \mathrm { y } = \mathrm { b }$ is tangent to the circle $x ^ { 2 } + y ^ { 2 } - 2 x - 2 y + 1 = 0$. Then $\mathrm { b } =$
(A) $-2$ or $12$
(B) $2$ or $-12$
(C) $-2$ or $-12$
(D) $2$ or $12$

8. Given that the line $l$: $x + a y - 1 = 0 ( a \in R )$ is an axis of symmetry of the circle $C$: $x ^ { 2 } + y ^ { 2 } - 4 x - 2 y + 1 = 0$. A tangent line to circle $C$ is drawn from point $\mathrm { A } ( - 4 , \mathrm { a } )$, with tangent point $B$. Then $| \mathrm { AB } | =$
A. $2$
B. $4 \sqrt { 2 }$
C. $6$
D. $2 \sqrt { 10 }$

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 $a < b$, $e_1 > e_2$

10. In the rectangular coordinate system $x O y$, among all circles with center at point $( 1,0 )$ and tangent to the line $m x - y - 2 m - 1 = 0 ( m \in R )$, the standard equation of the circle with the largest radius is $\_\_\_\_$.

10. Let line $l$ intersect the parabola $y ^ { 2 } = 4 x$ at points $A , B$, and be tangent to the circle $( x - 5 ) ^ { 2 } + y ^ { 2 } = r ^ { 2 } ( r > 0 )$ at point $M$. If $M$ is the midpoint of segment $A B$, and there are exactly 4 such lines $l$, then the range of $r$ is
(A) $( 1,3 )$
(B) $( 1,4 )$
(C) $( 2,3 )$
(D) $( 2,4 )$
II. Fill in the Blanks:

10. Let line $l$ intersect the parabola $y ^ { 2 } = 4 x$ at points $\mathrm { A }$ and $\mathrm { B }$, and be tangent to the circle $( x - 5 ) ^ { 2 } + y ^ { 2 } = r ^ { 2 }$ $(r > 0)$ at point $M$, where $M$ is the midpoint of segment $A B$. If there are exactly $4$ such lines $l$, then the range of $r$ is
(A) $( 1, 3 )$
(B) $( 1, 4 )$
(C) $( 2, 3 )$
(D) $( 2, 4 )$
II. Fill in the Blanks

If point $\mathrm { P } ( 1,2 )$ lies on a circle centered at the origin, then the equation of the tangent line to the circle at point $P$ is $\_\_\_\_$ .

13. If the line $3 x - 4 y + 5 = 0$ intersects the circle $x ^ { 2 } + y ^ { 2 } = r ^ { 2 } \quad ( r > 0 )$ at points $A$ and $B$, and $\angle A O B = 120 ^ { \circ }$ (where O is the coordinate origin), then $r =$ $\_\_\_\_$.

14. As shown in question (14), chords $\mathrm { AB }$ and $\mathrm { CD }$ of circle $O$ intersect at point $E$. A tangent line to circle $O$ is drawn through point $A$ and intersects the extension of $DC$ at point $P$. If $P A = 6 , A E = 9 , P C = 3 , C E : E D = 2 : 1$, then $B E = $ $\_\_\_\_$ . [Figure]

14. As shown in the figure, circle $C$ is tangent to the $x$-axis at point $T(1,0)$ and intersects the positive $y$-axis at two points $A$ and $B$ (with $B$ above $A$), and $|AB| = 2$. (I) The standard equation of circle $C$ is $\_\_\_\_$ ; (II) A line is drawn through point $A$ intersecting circle $O: x^2 + y^2 = 1$ at points $M$ and $N$. Consider the following three conclusions:
(1) $\frac{|NA|}{|NB|} = \frac{|MA|}{|MB|}$ ;
(2) $\frac{|NB|}{|NA|} - \frac{|MA|}{|MB|} = 2$ ;
(3) $\frac{|NB|}{|NA|} + \frac{|MA|}{|MB|} = 2\sqrt{2}$ .
The correct conclusion(s) is/are $\_\_\_\_$ . (Write the numbers of all correct conclusions)
(B) Optional Questions (Choose one of questions 15 and 16 to answer. First fill in the box after the question number you choose on the answer sheet with a 2B pencil. If you choose both, only question 15 will be graded.)

15. (Elective 4-1: Geometric Proof) As shown in the figure, $PA$ is tangent to the circle at point $A$, and $PBC$ is a secant line with $BC = 3PB$. Then $\frac{AB}{AC} = $ $\_\_\_\_$ .

15. For the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ , the right focus $\mathrm { F } ( c , 0 )$ is symmetric to point Q with respect to the line $y = \frac { b } { c } x$ , and Q lies on the ellipse. Then the eccentricity of the ellipse is $\_\_\_\_$. III. Solution Questions (This section contains 5 questions, 74 points total. Solutions should include explanations, proofs, or calculation steps.)

16. As shown in the figure, circle C is tangent to the x-axis at point $T ( 1,0 )$, and intersects the positive y-axis at two points $\mathrm { A } , \mathrm { B }$ (B is above A), with $| A B | = 2$.
(1) The standard equation of circle C is $\_\_\_\_$.
(2) The x-intercept of the tangent line to circle C at point B is $\_\_\_\_$. [Figure]

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]

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.

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]