You are given the following: a circle, one of its diameters $AB$ and a point $X$.
(a) Using only a straight-edge, show in the given figure how to draw a line perpendicular to $AB$ passing through $X$. No credit will be given without full justification. (Recall that a straight-edge is a ruler without any markings. Given two points, a straight-edge can be used to draw the line passing through the given points.)
(b) Do NOT draw any of your work for this part in the given figure. Reconsider your procedure to see if it can be made to work if the point $X$ is in some other position, e.g., when it is inside the circle or to the ``left/right'' of the circle. Clearly specify all positions of the point $X$ for which your procedure in part (a), or a small extension/variation of it, can be used to obtain the perpendicular to $AB$ through $X$. Justify your answer.

[7 points] Suppose $\mathrm{A}, \mathrm{B}, \mathrm{C}, \mathrm{D}$ are points on a circle such that AC and BD are diameters of that circle. Suppose $\mathrm{AB} = 12$ and $\mathrm{BC} = 5$. Let P be a point on the arc of the circle from A to B (the arc that does not contain points C and D). Let the distances of P from $\mathrm{A}, \mathrm{B}, \mathrm{C}$ and D be $a, b, c$ and $d$ respectively. Find the values of $\frac{a+b}{c+d}$ and $\frac{a-b}{d-c}$. You may assume $d \neq c$ so the second ratio makes sense.

As shown in the figure, a line passes through the focus F of the parabola $y ^ { 2 } = 12 x$ and meets the parabola at two points $\mathrm { A } , \mathrm { B }$. Let C and D be the feet of the perpendiculars from A and B to the directrix $l$ respectively. When $\overline { \mathrm { AC } } = 4$, what is the length of segment BD? [3 points]
(1) 12
(2) $\frac { 25 } { 2 }$
(3) 13
(4) $\frac { 27 } { 2 }$
(5) 14

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.)

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

16. (This question is worth 12 points) This question has three optional parts I, II, and III. Please select any two to answer and write your solutions in the corresponding answer areas on the answer sheet. If you answer all three, only the first two will be graded. I (This question is worth 6 points) Elective 4-1: Geometric Proof As shown in Figure 5, in circle O, two chords AB and CD intersect at point E, with midpoints M and N respectively. The line MO intersects line CD at point F. Prove: (I) $\angle \mathrm { MEN } + \angle \mathrm { NOM } = 180 ^ { \circ }$; (II) $\mathrm { FE } \cdot \mathrm { FN } = \mathrm { FM } \cdot \mathrm { FO }$
[Figure]
Figure 5
II. (This question is worth 6 points) Elective 4-4: Coordinate Systems and Parametric Equations Given the line $l: \left\{ \begin{array} { l } x = 5 + \frac { \sqrt { 3 } } { 2 } t \\ y = \sqrt { 3 } + \frac { 1 } { 2 } t \end{array} \right.$ (where t is the parameter). With the origin as the pole and the positive x-axis as the polar axis, the polar equation of curve C is $\rho = 2 \cos \theta$
(i) Convert the polar equation of curve C to rectangular coordinates; (II) Let the rectangular coordinates of point M be $( 5 , \sqrt { 3 } )$. The line $l$ intersects curve C at points $A$ and $B$. Find the value of $| M A | \cdot | M B |$ III. (This question is worth 6 points) Elective 4-5: Inequalities Let $\mathrm { a } > 0$, $\mathrm { b } > 0$, and $\mathrm { a } + \mathrm { b } = \frac { 1 } { a } + \frac { 1 } { b }$. Prove
(i) $\mathrm { a } + \mathrm { b } \geqslant 2$;
(ii) $\mathrm { a } ^ { 2 } + \mathrm { a } < 2$ and $\mathrm { b } ^ { 2 } + \mathrm { b } < 2$ cannot both be true.

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) $10$

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

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) $\frac{5}{2}$

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

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

Given the circle $x ^ { 2 } + y ^ { 2 } - 6 x = 0$ , the minimum length of the chord cut by this circle from a line passing through the point $( 1,2 )$ is
A. 1
B. 2
C. 3
D. 4

11. ACD

Solution: The radius of the circle is $r = 4$. The equation of line $AB$ is $y = - \frac { 1 } { 2 } x + 2$. Drawing a line through $P$ parallel to $AB$, the distance between the two lines is $d = \frac { \left| \frac { 15 } { 2 } - 2 \right| } { \sqrt { 1 + \left( - \frac { 1 } { 2 } \right) ^ { 2 } } } = \frac { 11 } { \sqrt { 5 } }$. Since $d < 6$, the maximum distance from $P$ to line $AB$ is $d + r$, so A is correct; the minimum distance from $P$ to line $AB$ is $d - r < 2$, so B is incorrect; the extremum of $\angle P A B$ is attained when $PB$ is tangent to the circle. We have $O B = \sqrt { 5 ^ { 2 } + 3 ^ { 2 } } = \sqrt { 34 }$. Since $PB$ is tangent to $OB$, we have $PB \perp O B$. By the Pythagorean theorem, $| P B | = \sqrt { 34 - 4 ^ { 2 } } = \sqrt { 18 }$. The two tangent lines from a point to a circle have equal length, so C and D are correct. The answer is $ACD$.
When $A _ { 1 } P \perp B P$, there are two points $P$ satisfying the condition, so C is incorrect. For option D, let $E$ be the midpoint of $C C _ { 1 }$, and let $G$ be the center of rectangle $A A _ { 1 } B _ { 1 } B$. When $\mu = \frac { 1 } { 2 }$, $P$ is a point on $EF$. Since $E G \perp$ plane $A A _ { 1 } B _ { 1 }$ and $A _ { 1 } B \perp A B _ { 1 }$, we have $A _ { 1 } B \perp$ plane $E A B _ { 1 }$. When $P$ coincides with $E$, the condition is satisfied. There is a unique plane perpendicular to line $A _ { 1 } B$ passing through such points.

III. Fill in the Blank Questions

13. 1
Solution: Setting $f ( x ) = f ( - x )$ gives $x ^ { 3 } \left( 2 ^ { x } + 2 ^ { - x } \right) ( a - 1 ) = 0$ for all $x$, so $a = 1$.

16. Given an ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ ( $a > b > 0$ ), with upper vertex $A$, two foci $F _ { 1 }$ and $F _ { 2 }$, and eccentricity $\frac { 1 } { 2 }$. A line through $F _ { 1 }$ perpendicular to $A F _ { 2 }$ intersects $C$ at points $D$ and $E$, with $| D E | = 6$. The perimeter of $\triangle A D E$ is $\_\_\_\_$ .
IV. Solution Questions: This section contains 6 questions, for a total of 70 points. Solutions should include explanations, proofs, or calculation steps.

The eccentricity of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \ (a > 0 , b > 0)$ is $\sqrt{5}$ . One of its asymptotes intersects the circle $(x - 2)^{2} + (y - 3)^{2} = 1$ at points $A , B$ , then $|AB| =$
A. $\frac{1}{5}$
B. $\frac{\sqrt{5}}{5}$
C. $\frac{2\sqrt{5}}{5}$
D. $\frac{4\sqrt{5}}{5}$

128. Quadrilateral $ABCD$ is inscribed in a circle. If $AB$ is the farthest chord and $BC$ is the closest chord to the center of this circle, which relationship between the angles cannot hold?

• [(1)] $\hat{D} > \hat{C}$
• [(2)] $\hat{B} > \hat{C}$
• [(3)] $\hat{A} > \hat{B}$
• [(4)] $\hat{B} > \hat{D}$

130. Two circles with radii $4$ and $8$ are internally tangent at point $A$. A chord $BC$ of the large circle is tangent to the small circle, and the line through the center of the small circle parallel to the radical axis passes through point $P$. What is $PB \times PC$?
(1) $24$ (2) $32$ (3) $36$ (4) $48$
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133- The common chord of circle $C$ with equation $x^2 + y^2 - 4x + y^2 = 6$ is tangent to the first region of circle $C$. If the point $(-1, 4)$ lies on it, the equation of the common chord is which of the following?
(1) $x^2 + y^2 - y + 3x = 6$ (2) $x^2 + y^2 + 3y - x = 6$
(3) $x^2 + y^2 - 2y + x = 6$ (4) $x^2 + y^2 - 3y - x = 6$

131- In the figure below, segment $AC$ equals chord $AB$. Which of the following is necessarily true?
[Figure: Circle with points A, B, C, D where AC is a chord equal to chord AB]

• [(1)] $BC = BA$
• [(2)] $BD = AC$
• [(3)] $BC = BD$
• [(4)] $DA = DC$

36. Line $d$ has equation $y - x = 0$. A circle with center at the origin has a radius twice that of another circle. If line $d$ is tangent to the smaller circle with equation $x^2 + y^2 + 6x - 2y = r$, what is the product of the lengths of the chord(s) of intersection of the two circles?
(1) $\dfrac{5}{2}$ (2) $\dfrac{5}{4}$ (3) $\dfrac{65}{32}$ (4) $\dfrac{65}{64}$

Consider a circle with centre $O$. Two chords $AB$ and $CD$ extended intersect at a point $P$ outside the circle. If $\angle AOC = 43^\circ$ and $\angle BPD = 18^\circ$, then the value of $\angle BOD$ is
(A) $36^\circ$
(B) $29^\circ$
(C) $7^\circ$
(D) $25^\circ$

Consider a circle with centre $O$. Two chords $AB$ and $CD$ extended intersect at a point $P$ outside the circle. If $\angle AOC = 43^\circ$ and $\angle BPD = 18^\circ$, then the value of $\angle BOD$ is
(A) $36^\circ$
(B) $29^\circ$
(C) $7^\circ$
(D) $25^\circ$

Consider a circle with centre $O$. Two chords $A B$ and $C D$ extended intersect at a point $P$ outside the circle. If $\angle A O C = 43 ^ { \circ }$ and $\angle B P D = 18 ^ { \circ }$, then the value of $\angle B O D$ is
(A) $36 ^ { \circ }$
(B) $29 ^ { \circ }$
(C) $7 ^ { \circ }$
(D) $25 ^ { \circ }$

Consider a circle of radius 6 as given in the diagram below. Let $B$, $C , D$ and $E$ be points on the circle such that $B D$ and $C E$, when extended, intersect at $A$. If $A D$ and $A E$ have length 5 and 4 respectively, and $D B C$ is a right angle, then show that the length of $B C$ is $\frac { 12 + 9 \sqrt { 15 } } { 5 }$.