QUESTION 155
The area of a circle with radius 5 cm is
(A) $10\pi$ cm$^2$
(B) $15\pi$ cm$^2$
(C) $20\pi$ cm$^2$
(D) $25\pi$ cm$^2$
(E) $30\pi$ cm$^2$

A waiter needs to choose a tray with a rectangular base to serve four glasses of sparkling wine that need to be arranged in a single row, parallel to the longer side of the tray, and with their bases completely supported on the tray. The base and upper edge of the glasses are circles with radius 4 cm and 5 cm, respectively.
The tray to be chosen should have a minimum area, in square centimeters, equal to
(A) 192.
(B) 300.
(C) 304.
(D) 320.
(E) 400.

Let $C _ { 1 } , C _ { 2 }$ be two circles of equal radii $R$. If $C _ { 1 }$ passes through the centre of $C _ { 2 }$ prove that the area of the region common to them is $\frac { R ^ { 2 } } { 6 } ( 4 \pi - \sqrt { 27 } )$.

As shown in the figure, draw a circle $\mathrm { O } _ { 1 }$ centered at the origin with radius 3, and let the four points where circle $\mathrm { O } _ { 1 }$ meets the coordinate axes be $\mathrm { A } _ { 1 } ( 0,3 )$, $\mathrm { B } _ { 1 } ( - 3,0 ) , \mathrm { C } _ { 1 } ( 0 , - 3 ) , \mathrm { D } _ { 1 } ( 3,0 )$ respectively. Two circles passing through both points $\mathrm { B } _ { 1 }$ and $\mathrm { D } _ { 1 }$ and centered at points $\mathrm { A } _ { 1 }$ and $\mathrm { C } _ { 1 }$ respectively meet the $y$-axis inside circle $\mathrm { O } _ { 1 }$ at points $\mathrm { C } _ { 2 }$ and $\mathrm { A } _ { 2 }$ respectively. Let $S _ { 1 }$ be the area of the region enclosed by arc $\mathrm { B } _ { 1 } \mathrm { A } _ { 1 } \mathrm { D } _ { 1 }$ and arc $\mathrm { B } _ { 1 } \mathrm { A } _ { 2 } \mathrm { D } _ { 1 }$, and let $T _ { 1 }$ be the area of the region enclosed by arc $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ and arc $\mathrm { B } _ { 1 } \mathrm { C } _ { 2 } \mathrm { D } _ { 1 }$. Draw circle $\mathrm { O } _ { 2 }$ with diameter $\mathrm { A } _ { 2 } \mathrm { C } _ { 2 }$, and let the two points where circle $\mathrm { O } _ { 2 }$ meets the $x$-axis be $\mathrm { B } _ { 2 }$ and $\mathrm { D } _ { 2 }$ respectively. Two circles passing through both points $\mathrm { B } _ { 2 }$ and $\mathrm { D } _ { 2 }$ and centered at points $\mathrm { A } _ { 2 }$ and $\mathrm { C } _ { 2 }$ respectively meet the $y$-axis inside circle $\mathrm { O } _ { 2 }$ at points $\mathrm { C } _ { 3 }$ and $\mathrm { A } _ { 3 }$ respectively. Let $S _ { 2 }$ be the area of the region enclosed by arc $\mathrm { B } _ { 2 } \mathrm { A } _ { 2 } \mathrm { D } _ { 2 }$ and arc $\mathrm { B } _ { 2 } \mathrm { A } _ { 3 } \mathrm { D } _ { 2 }$, and let $T _ { 2 }$ be the area of the region enclosed by arc $\mathrm { B } _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ and arc $\mathrm { B } _ { 2 } \mathrm { C } _ { 3 } \mathrm { D } _ { 2 }$. Continuing this process, let $S _ { n }$ be the area of the region enclosed by arc $\mathrm { B } _ { n } \mathrm { A } _ { n } \mathrm { D } _ { n }$ and arc $\mathrm { B } _ { n } \mathrm { A } _ { n + 1 } \mathrm { D } _ { n }$, and let $T _ { n }$ be the area of the region enclosed by arc $\mathrm { B } _ { n } \mathrm { C } _ { n } \mathrm { D } _ { n }$ and arc $\mathrm { B } _ { n } \mathrm { C } _ { n + 1 } \mathrm { D } _ { n }$. What is the value of $\sum _ { n = 1 } ^ { \infty } \left( S _ { n } + T _ { n } \right)$? [4 points]
(1) $6 ( \sqrt { 2 } + 1 )$
(2) $6 ( \sqrt { 3 } + 1 )$
(3) $6 ( \sqrt { 5 } + 1 )$
(4) $9 ( \sqrt { 2 } + 1 )$
(5) $9 ( \sqrt { 3 } + 1 )$

As shown in the figure, draw a circle $\mathrm { O } _ { 1 }$ centered at the origin with radius 3, and let the four points where circle $\mathrm { O } _ { 1 }$ meets the coordinate axes be $\mathrm { A } _ { 1 } ( 0,3 )$, $\mathrm { B } _ { 1 } ( - 3,0 ) , \mathrm { C } _ { 1 } ( 0 , - 3 ) , \mathrm { D } _ { 1 } ( 3,0 )$ respectively. Two circles passing through both points $\mathrm { B } _ { 1 } , \mathrm { D } _ { 1 }$ and centered at points $\mathrm { A } _ { 1 } , \mathrm { C } _ { 1 }$ respectively meet the $y$-axis inside circle $\mathrm { O } _ { 1 }$ at points $\mathrm { C } _ { 2 } , \mathrm {~A} _ { 2 }$ respectively.
Let $S _ { 1 }$ be the area of the region enclosed by arc $\mathrm { B } _ { 1 } \mathrm {~A} _ { 1 } \mathrm { D } _ { 1 }$ and arc $\mathrm { B } _ { 1 } \mathrm {~A} _ { 2 } \mathrm { D } _ { 1 }$, and let $T _ { 1 }$ be the area of the region enclosed by arc $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$ and arc $\mathrm { B } _ { 1 } \mathrm { C } _ { 2 } \mathrm { D } _ { 1 }$.
Draw circle $\mathrm { O } _ { 2 }$ with segment $\mathrm { A } _ { 2 } \mathrm { C } _ { 2 }$ as diameter, and let the two points where circle $\mathrm { O } _ { 2 }$ meets the $x$-axis be $\mathrm { B } _ { 2 } , \mathrm { D } _ { 2 }$ respectively. Two circles passing through both points $\mathrm { B } _ { 2 } , \mathrm { D } _ { 2 }$ and centered at points $\mathrm { A } _ { 2 } , \mathrm { C } _ { 2 }$ respectively meet the $y$-axis inside circle $\mathrm { O } _ { 2 }$ at points $\mathrm { C } _ { 3 } , \mathrm {~A} _ { 3 }$ respectively.
Let $S _ { 2 }$ be the area of the region enclosed by arc $\mathrm { B } _ { 2 } \mathrm {~A} _ { 2 } \mathrm { D } _ { 2 }$ and arc $\mathrm { B } _ { 2 } \mathrm {~A} _ { 3 } \mathrm { D } _ { 2 }$, and let $T _ { 2 }$ be the area of the region enclosed by arc $\mathrm { B } _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ and arc $\mathrm { B } _ { 2 } \mathrm { C } _ { 3 } \mathrm { D } _ { 2 }$.
Continuing this process, let $S _ { n }$ be the area of the region enclosed by arc $\mathrm { B } _ { n } \mathrm {~A} _ { n } \mathrm { D } _ { n }$ and arc $\mathrm { B } _ { n } \mathrm {~A} _ { n + 1 } \mathrm { D } _ { n }$ obtained in the $n$-th step, and let $T _ { n }$ be the area of the region enclosed by arc $\mathrm { B } _ { n } \mathrm { C } _ { n } \mathrm { D } _ { n }$ and arc $\mathrm { B } _ { n } \mathrm { C } _ { n + 1 } \mathrm { D } _ { n }$. What is the value of $\sum _ { n = 1 } ^ { \infty } \left( S _ { n } + T _ { n } \right)$? [4 points]
(1) $6 ( \sqrt { 2 } + 1 )$
(2) $6 ( \sqrt { 3 } + 1 )$
(3) $6 ( \sqrt { 5 } + 1 )$
(4) $9 ( \sqrt { 2 } + 1 )$
(5) $9 ( \sqrt { 3 } + 1 )$

As shown in the figure, there are two circular disks with distance between centers $\sqrt { 3 }$ and radius 1, and a plane $\alpha$. The line $l$ passing through the centers of each disk is perpendicular to the planes of the two disks and makes an angle of $60 ^ { \circ }$ with plane $\alpha$. When sunlight shines perpendicular to plane $\alpha$ as shown in the figure, what is the area of the shadow cast by the two disks on plane $\alpha$? (Note: the thickness of the disks is negligible.) [4 points]
(1) $\frac { \sqrt { 3 } } { 3 } \pi + \frac { 3 } { 8 }$
(2) $\frac { 2 } { 3 } \pi + \frac { \sqrt { 3 } } { 4 }$
(3) $\frac { 2 \sqrt { 3 } } { 3 } \pi + \frac { 1 } { 8 }$
(4) $\frac { 4 } { 3 } \pi + \frac { \sqrt { 3 } } { 16 }$
(5) $\frac { 2 \sqrt { 3 } } { 3 } \pi + \frac { 3 } { 4 }$

As shown in the figure, in the coordinate plane, for two points $\mathrm { A } , \mathrm { B }$ on the $x$-axis, the parabola $p _ { 1 }$ with vertex at A and the parabola $p _ { 2 }$ with vertex at B satisfy the following conditions. What is the area of triangle ABC? [4 points] (가) The focus of $p _ { 1 }$ is B, and the focus of $p _ { 2 }$ is the origin O. (나) $p _ { 1 }$ and $p _ { 2 }$ meet at two points $\mathrm { C } , \mathrm { D }$ on the $y$-axis. (다) $\overline { \mathrm { AB } } = 2$
(1) $4 ( \sqrt { 2 } - 1 )$
(2) $3 ( \sqrt { 3 } - 1 )$
(3) $2 ( \sqrt { 5 } - 1 )$
(4) $\sqrt { 3 } + 1$
(5) $\sqrt { 5 } + 1$

For the parabola $y ^ { 2 } = 4 x$, let $l$ be the tangent line at point $\mathrm { A } ( 4,4 )$. Let B be the intersection of line $l$ and the directrix of the parabola, C be the intersection of line $l$ and the $x$-axis, and D be the intersection of the directrix and the $x$-axis. What is the area of triangle BCD? [3 points]
(1) $\frac { 7 } { 4 }$
(2) 2
(3) $\frac { 9 } { 4 }$
(4) $\frac { 5 } { 2 }$
(5) $\frac { 11 } { 4 }$

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. $\frac { | B F | ^ { 2 } + 1 } { | A F | ^ { 2 } + 1 }$

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. (12 points)
Let $F _ { 1 } , F _ { 2 }$ be the two foci of the ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$, $P$ be a point on $C$, and $O$ be the origin.
(1) If $\triangle P O F _ { 2 }$ is an equilateral triangle, find the eccentricity of $C$;
(2) If there exists a point $P$ such that $P F _ { 1 } \perp P F _ { 2 }$ and the area of $\triangle F _ { 1 } P F _ { 2 }$ equals 16, find the value of $b$ and the range of values for $a$.

16. Let $F _ { 1 } , F _ { 2 }$ be the two foci of the ellipse $C : \frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1$. Let $P , Q$ be two points on $C$ that are symmetric with respect to the origin, and $| P Q | = \left| F _ { 1 } F _ { 2 } \right|$. Then the area of quadrilateral $P F _ { 1 } Q F _ { 2 }$ is $\_\_\_\_$ .

III. Solution Questions: Total 70 points. Solutions should include explanations, proofs, or calculation steps. Questions 17--21 are required questions that all students must answer. Questions 22 and 23 are optional questions; students should answer according to the requirements.

(A) Required Questions: Total 60 points.

15. Let $F_1, F_2$ be the two foci of the ellipse $C: \frac{x^2}{16} + \frac{y^2}{4} = 1$. Let $P, Q$ be two points on $C$ that are symmetric about the origin, and $|PQ| = |F_1F_2|$. Then the area of quadrilateral $PF_1QF_2$ is $\_\_\_\_$. [Figure]

Given the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ with eccentricity $\frac{\sqrt{2}}{2}$ and major axis length 4.
(1) Find the equation of $C$;
(2) A line $l$ passing through the point $(0, -2)$ intersects $C$ at points $A, B$. Let $O$ be the origin. If the area of $\triangle OAB$ is $\sqrt{2}$, find $|AB|$.

Let $n$ be a positive integer. Consider a square $S$ of side $2n$ units. Divide $S$ into $4n^2$ unit squares by drawing $2n - 1$ horizontal and $2n - 1$ vertical lines one unit apart. A circle of diameter $2n - 1$ is drawn with its centre at the intersection of the two diagonals of the square $S$. How many of these unit squares contain a portion of the circumference of the circle?
(A) $4n - 2$
(B) $4n$
(C) $8n - 4$
(D) $8n - 2$

Let $n$ be a positive integer. Consider a square $S$ of side $2n$ units with sides parallel to the coordinate axes. Divide $S$ into $4n^2$ unit squares by drawing $2n-1$ horizontal and $2n-1$ vertical lines one unit apart. A circle of diameter $2n-1$ is drawn with its centre at the intersection of the two diagonals of the square $S$. How many of these unit squares contain a portion of the circumference of the circle?
(A) $4n - 2$
(B) $4n$
(C) $8n - 4$
(D) $8n - 2$

Let $n$ be a positive integer. Consider a square $S$ of side $2n$ units with sides parallel to the coordinate axes. Divide $S$ into $4 n ^ { 2 }$ unit squares by drawing $2n - 1$ horizontal and $2n - 1$ vertical lines one unit apart. A circle of diameter $2n - 1$ is drawn with its centre at the intersection of the two diagonals of the square $S$. How many of these unit squares contain a portion of the circumference of the circle?
(A) $4 n - 2$
(B) $4 n$
(C) $8 n - 4$
(D) $8 n - 2$

Suppose that $P Q$ and $R S$ are two chords of a circle intersecting at a point $O$. It is given that $P O = 3 \mathrm {~cm}$ and $S O = 4 \mathrm {~cm}$. Moreover, the area of the triangle $P O R$ is $7 \mathrm {~cm} ^ { 2 }$. Find the area of the triangle $Q O S$.

Two vertices of a square lie on a circle of radius $r$ and the other two vertices lie on a tangent to this circle. Then the length of the side of the square is
(A) $\frac { 3 r } { 2 }$
(B) $\frac { 4 r } { 3 }$
(C) $\frac { 6 r } { 5 }$
(D) $\frac { 8 r } { 5 }$.

Consider the curves $x ^ { 2 } + y ^ { 2 } - 4 x - 6 y - 12 = 0,9 x ^ { 2 } + 4 y ^ { 2 } - 900 = 0$ and $y ^ { 2 } - 6 y - 6 x + 51 = 0$. The maximum number of disjoint regions into which these curves divide the $XY$-plane (excluding the curves themselves), is
(A) 4 .
(B) 5 .
(C) 6 .
(D) 7 .

Consider a triangle with vertices $( 0,0 ) , ( 1,2 )$ and $( - 4,2 )$. Let $A$ be the area of the triangle and $B$ be the area of the circumcircle of the triangle. Then $\frac { B } { A }$ equals
(A) $\frac { \pi } { 2 }$.
(B) $\frac { 5 \pi } { 4 }$.
(C) $\frac { 3 } { \sqrt { 2 } } \pi$.
(D) $2 \pi$.

In the following figure, $O A B$ is a quarter-circle. The unshaded region is a circle to which $O A$ and $C D$ are tangents. If $C D$ is of length 10 and is parallel to $O A$, then the area of the shaded region in the above figure equals
(A) $25 \pi$.
(B) $50 \pi$.
(C) $75 \pi$.
(D) $100 \pi$.

In the adjoining figure, $C$ is the centre of the circle drawn, $A, F, E$ lie on the circle and $BCDF$ is a rectangle. If $\frac{DE}{AB} = 2$, then $\frac{FE}{FA}$ equals
(A) $\sqrt{\frac{3}{2}}$
(B) $\sqrt{2}$
(C) $\sqrt{\frac{5}{2}}$
(D) $\sqrt{3}$

The common tangents to the circle $x^2 + y^2 = 2$ and the parabola $y^2 = 8x$ touch the circle at the points $P, Q$ and the parabola at the points $R, S$. Then the area of the quadrilateral $PQRS$ is
(A) 3
(B) 6
(C) 9
(D) 15

Consider the ellipse
$$\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 1$$
Let $H ( \alpha , 0 ) , 0 < \alpha < 2$, be a point. A straight line drawn through $H$ parallel to the $y$-axis crosses the ellipse and its auxiliary circle at points $E$ and $F$ respectively, in the first quadrant. The tangent to the ellipse at the point $E$ intersects the positive $x$-axis at a point $G$. Suppose the straight line joining $F$ and the origin makes an angle $\phi$ with the positive $x$-axis.
List-I (I) If $\phi = \frac { \pi } { 4 }$, then the area of the triangle $F G H$ is (II) If $\phi = \frac { \pi } { 3 }$, then the area of the triangle $F G H$ is (III) If $\phi = \frac { \pi } { 6 }$, then the area of the triangle $F G H$ is (IV) If $\phi = \frac { \pi } { 12 }$, then the area of the triangle $F G H$ is
List-II (P) $\frac { ( \sqrt { 3 } - 1 ) ^ { 4 } } { 8 }$ (Q) 1 (R) $\frac { 3 } { 4 }$ (S) $\frac { 1 } { 2 \sqrt { 3 } }$ (T) $\frac { 3 \sqrt { 3 } } { 2 }$
The correct option is:
(A) (I) → (R); (II) → (S); (III) → (Q); (IV) → (P)
(B) (I) → (R); (II) → (T); (III) → (S); (IV) → (P)
(C) (I) → (Q); (II) → (T); (III) → (S); (IV) → (P)
(D) (I) → (Q); (II) → (S); (III) → (Q); (IV) → (P)