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

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) $10$

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. $\frac{4\sqrt{5}}{5}$

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 $\frac { 12 + 9 \sqrt { 15 } } { 5 }$.

The angle subtended at the origin by the common chord of the circles $x^2 + y^2 - 6x - 6y = 0$ and $x^2 + y^2 = 36$ is
(A) $\pi/2$
(B) $\pi/4$
(C) $\pi/3$
(D) $2\pi/3$

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) $25 ^ { \circ }$.

A straight line through the vertex $P$ of a triangle $P Q R$ intersects the side $Q R$ at the point $S$ and the circumcircle of the triangle $P Q R$ at the point $T$. If $S$ is not the centre of the circumcircle, then
(A) $\frac { 1 } { P S } + \frac { 1 } { S T } < \frac { 2 } { \sqrt { Q S \times S R } }$
(B) $\frac { 1 } { P S } + \frac { 1 } { S T } > \frac { 2 } { \sqrt { Q S \times S R } }$
(C) $\frac { 1 } { P S } + \frac { 1 } { S T } < \frac { 4 } { Q R }$
(D) $\frac { 1 } { P S } + \frac { 1 } { S T } > \frac { 4 } { Q R }$

Consider
$$\begin{aligned} & L _ { 1 } : 2 x + 3 y + p - 3 = 0 \\ & L _ { 2 } : 2 x + 3 y + p + 3 = 0 \end{aligned}$$
where $p$ is a real number, and $C : x ^ { 2 } + y ^ { 2 } + 6 x - 10 y + 30 = 0$. STATEMENT-1 : If line $L _ { 1 }$ is a chord of circle $C$, then line $L _ { 2 }$ is not always a diameter of circle $C$.
and
STATEMENT-2 : If line $L _ { 1 }$ is a diameter of circle $C$, then line $L _ { 2 }$ is not a chord of circle $C$.
(A) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
(B) STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
(C) STATEMENT-1 is True, STATEMENT-2 is False
(D) STATEMENT-1 is False, STATEMENT-2 is True

Let the curve $C$ be the mirror image of the parabola $y ^ { 2 } = 4 x$ with respect to the line $x + y + 4 = 0$. If $A$ and $B$ are the points of intersection of $C$ with the line $y = - 5$, then the distance between $A$ and $B$ is

If a chord, which is not a tangent, of the parabola $y^2 = 16x$ has the equation $2x + y = p$, and midpoint $(h, k)$, then which of the following is(are) possible value(s) of $p$, $h$ and $k$?
[A] $p = -2, h = 2, k = -4$
[B] $p = -1, h = 1, k = -3$
[C] $p = 2, h = 3, k = -4$
[D] $p = 5, h = 4, k = -3$

A line $y = m x + 1$ intersects the circle $( x - 3 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = 25$ at the points $P$ and $Q$. If the midpoint of the line segment $P Q$ has $x$-coordinate $- \frac { 3 } { 5 }$, then which one of the following options is correct?
(A) $\quad - 3 \leq m < - 1$
(B) $2 \leq m < 4$
(C) $4 \leq m < 6$
(D) $6 \leq m < 8$