In the coordinate plane, for point $\mathrm { A } ( 0,4 )$ and point P on the ellipse $\frac { x ^ { 2 } } { 5 } + y ^ { 2 } = 1$, let Q be the point other than A among the two points where the line passing through A and P meets the circle $x ^ { 2 } + ( y - 3 ) ^ { 2 } = 1$. When point P passes through all points on the ellipse, what is the length of the figure traced by point Q? [3 points]
(1) $\frac { \pi } { 6 }$
(2) $\frac { \pi } { 4 }$
(3) $\frac { \pi } { 3 }$
(4) $\frac { 2 } { 3 } \pi$
(5) $\frac { 3 } { 4 } \pi$

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 a rhombus ABCD with side length 10, an ellipse with diagonal BD as the major axis and diagonal AC as the minor axis has a distance between the two foci of $10 \sqrt { 2 }$. What is the area of rhombus ABCD? [3 points]
(1) $55 \sqrt { 3 }$
(2) $65 \sqrt { 2 }$
(3) $50 \sqrt { 3 }$
(4) $45 \sqrt { 3 }$
(5) $45 \sqrt { 2 }$

There is a circle with radius 1. As shown in the figure, a rectangle with the ratio of width to height of $3 : 1$ is inscribed in this circle, and the common part of the interior of the circle and the exterior of the rectangle is colored to obtain the figure $R _ { 1 }$. In figure $R _ { 1 }$, 2 circles are drawn tangent to three sides of the rectangle. A rectangle is drawn in each of the newly drawn circles using the same method as obtaining figure $R _ { 1 }$, and the colored figure obtained is $R _ { 2 }$. In figure $R _ { 2 }$, 4 circles are drawn tangent to three sides of the newly drawn rectangle. A rectangle is drawn in each of the newly drawn circles using the same method as obtaining figure $R _ { 1 }$, and the colored figure obtained is $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the area of the colored part in the $n$-th obtained figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 5 } { 4 } \pi - \frac { 5 } { 3 }$
(2) $\frac { 5 } { 4 } \pi - \frac { 3 } { 2 }$
(3) $\frac { 4 } { 3 } \pi - \frac { 8 } { 5 }$
(4) $\frac { 5 } { 4 } \pi - 1$
(5) $\frac { 4 } { 3 } \pi - \frac { 16 } { 15 }$

There is a circle with radius 1. As shown in the figure, a rectangle with a ratio of horizontal length to vertical length of $3 : 1$ is inscribed in this circle, and the common part of the interior of the circle and the exterior of the rectangle is colored to obtain a figure $R _ { 1 }$. In figure $R _ { 1 }$, 2 circles are drawn tangent to three sides of the rectangle. A rectangle is drawn in each of the newly drawn circles using the same method as obtaining figure $R _ { 1 }$, and the colored figure obtained is $R _ { 2 }$. In figure $R _ { 2 }$, 4 circles are drawn tangent to three sides of the newly drawn rectangles. A rectangle is drawn in each of the newly drawn circles using the same method as obtaining figure $R _ { 1 }$, and the colored figure obtained is $R _ { 3 }$. Continuing this process, let $S _ { n }$ be the area of the colored part in the $n$-th obtained figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 5 } { 4 } \pi - \frac { 5 } { 3 }$
(2) $\frac { 5 } { 4 } \pi - \frac { 3 } { 2 }$
(3) $\frac { 4 } { 3 } \pi - \frac { 8 } { 5 }$
(4) $\frac { 5 } { 4 } \pi - 1$
(5) $\frac { 4 } { 3 } \pi - \frac { 16 } { 15 }$

As shown in the figure, a cylinder with base radius 7 and a cone with base radius 5 and height 12 are placed on a plane $\alpha$, and the circumference of the base of the cone is inscribed in the circumference of the base of the cylinder. Let O be the center of the base of the cylinder that meets plane $\alpha$, and let A be the apex of the cone. A sphere $S$ with center B and radius 4 satisfies the following conditions. (가) The sphere $S$ is tangent to both the cylinder and the cone. (나) When $\mathrm { A } ^ { \prime }$ and $\mathrm { B } ^ { \prime }$ are the orthogonal projections of points $\mathrm { A }$ and $\mathrm { B }$ onto plane $\alpha$ respectively, $\angle \mathrm { A } ^ { \prime } \mathrm { OB } ^ { \prime } = 180 ^ { \circ }$.
When the acute angle between line AB and plane $\alpha$ is $\theta$, $\tan \theta = p$. Find the value of $100 p$. (Note: The center of the base of the cone and point $\mathrm { A } ^ { \prime }$ coincide.) [4 points]

In the coordinate plane, two lines $l _ { 1 } , l _ { 2 }$ tangent to the parabola $y ^ { 2 } = 8 x$ have slopes $m _ { 1 } , m _ { 2 }$ respectively. When $m _ { 1 } , m _ { 2 }$ are the two distinct roots of the equation $2 x ^ { 2 } - 3 x + 1 = 0$, what is the $x$-coordinate of the intersection point of $l _ { 1 }$ and $l _ { 2 }$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

In rectangle $\mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } \mathrm { C } _ { 1 } \mathrm { D } _ { 1 }$, $\overline { \mathrm { A } _ { 1 } \mathrm {~B} _ { 1 } } = 1 , \overline { \mathrm {~A} _ { 1 } \mathrm { D } _ { 1 } } = 2$. As shown in the figure, let $\mathrm { M } _ { 1 } , \mathrm {~N} _ { 1 }$ be the midpoints of segments $\mathrm { A } _ { 1 } \mathrm { D } _ { 1 }$ and $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$ respectively.
Draw a circular sector $\mathrm { N } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ with center $\mathrm { N } _ { 1 }$, radius $\overline { \mathrm { B } _ { 1 } \mathrm {~N} _ { 1 } }$, and central angle $\frac { \pi } { 2 }$, and draw a circular sector $\mathrm { D } _ { 1 } \mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ with center $\mathrm { D } _ { 1 }$, radius $\overline { \mathrm { C } _ { 1 } \mathrm { D } _ { 1 } }$, and central angle $\frac { \pi } { 2 }$.
The region enclosed by the arc $\mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ and segment $\mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ of sector $\mathrm { N } _ { 1 } \mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$ and the region enclosed by the arc $\mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ and segment $\mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ of sector $\mathrm { D } _ { 1 } \mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$ are shaded to form a checkmark shape, and the resulting figure is called $R _ { 1 }$.
In figure $R _ { 1 }$, a rectangle $\mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ is drawn with vertices at point $\mathrm { A } _ { 2 }$ on segment $\mathrm { M } _ { 1 } \mathrm {~B} _ { 1 }$, point $\mathrm { D } _ { 2 }$ on arc $\mathrm { M } _ { 1 } \mathrm { C } _ { 1 }$, and two points $\mathrm { B } _ { 2 } , \mathrm { C } _ { 2 }$ on side $\mathrm { B } _ { 1 } \mathrm { C } _ { 1 }$ such that $\overline { \mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } } : \overline { \mathrm { A } _ { 2 } \mathrm { D } _ { 2 } } = 1 : 2$. A checkmark shape is shaded in rectangle $\mathrm { A } _ { 2 } \mathrm {~B} _ { 2 } \mathrm { C } _ { 2 } \mathrm { D } _ { 2 }$ using the same method as for figure $R _ { 1 }$, and the resulting figure is called $R _ { 2 }$.
Continuing this process, let $S _ { n }$ be the area of the shaded region in the $n$-th figure $R _ { n }$. What is the value of $\lim _ { n \rightarrow \infty } S _ { n }$? [4 points]
(1) $\frac { 25 } { 19 } \left( \frac { \pi } { 2 } - 1 \right)$
(2) $\frac { 5 } { 4 } \left( \frac { \pi } { 2 } - 1 \right)$
(3) $\frac { 25 } { 21 } \left( \frac { \pi } { 2 } - 1 \right)$
(4) $\frac { 25 } { 22 } \left( \frac { \pi } { 2 } - 1 \right)$
(5) $\frac { 25 } { 23 } \left( \frac { \pi } { 2 } - 1 \right)$

In coordinate space, a sphere $S$ with center coordinates all positive has center at $(x, y, z)$ where $x > 0, y > 0, z > 0$, is tangent to the $x$-axis and $y$-axis respectively, and intersects the $z$-axis at two distinct points. The area of the circle formed by the intersection of sphere $S$ and the $xy$-plane is $64 \pi$, and the distance between the two intersection points with the $z$-axis is 8. What is the radius of sphere $S$? [4 points]
(1) 11
(2) 12
(3) 13
(4) 14
(5) 15

As shown in the figure, there is a point $\mathrm { A } ( 0 , a )$ on the $y$-axis and a point P moving on the ellipse $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 9 } = 1$ with foci $\mathrm { F } , \mathrm { F } ^ { \prime }$. When the minimum value of $\overline { \mathrm { AP } } - \overline { \mathrm { FP } }$ is 1, find the value of $a ^ { 2 }$. [4 points]

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) $\frac { 11 } { 4 }$

For two positive numbers $k , p$, two tangent lines are drawn from point $\mathrm { A } ( - k , 0 )$ to the parabola $y ^ { 2 } = 4 p x$. Let $\mathrm { F } , \mathrm { F } ^ { \prime }$ be the two points where these tangent lines meet the $y$-axis, and let $\mathrm { P } , \mathrm { Q }$ be the two points where they meet the parabola. When $\angle \mathrm { PAQ } = \frac { \pi } { 3 }$, if the length of the major axis of the ellipse with foci at $\mathrm { F } , \mathrm { F } ^ { \prime }$ and passing through points $\mathrm { P } , \mathrm { Q }$ is $4 \sqrt { 3 } + 12$, what is the value of $k + p$? [4 points]
(1) 8
(2) 10
(3) 12
(4) 14
(5) 16

In coordinate space, find the sum of all real numbers $k$ such that the plane $x + 8 y - 4 z + k = 0$ is tangent to the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 2 y - 3 = 0$. [3 points]

For a point P on the parabola $y ^ { 2 } = 12 x$ with focus F, when $\overline { \mathrm { PF } } = 9$, what is the $x$-coordinate of point P? [3 points]
(1) 6
(2) $\frac { 13 } { 2 }$
(3) 7
(4) $\frac { 15 } { 2 }$
(5) 8

There is an ellipse $\frac { x ^ { 2 } } { 49 } + \frac { y ^ { 2 } } { 33 } = 1$ with foci $\mathrm { F } , \mathrm { F } ^ { \prime }$. For a point P on the circle $x ^ { 2 } + ( y - 3 ) ^ { 2 } = 4$, let Q be the point with positive $y$-coordinate among the points where the line $\mathrm { F } ^ { \prime } \mathrm { P }$ meets this ellipse. Find the maximum value of $\overline { \mathrm { PQ } } + \overline { \mathrm { FQ } }$. [4 points]

There are two circles $C _ { 1 } , C _ { 2 }$ with centers $\mathrm { O } _ { 1 } , \mathrm { O } _ { 2 }$ respectively and radii equal to $\overline { \mathrm { O } _ { 1 } \mathrm { O } _ { 2 } }$. As shown in the figure, three distinct points $\mathrm { A } , \mathrm { B } , \mathrm { C }$ on circle $C _ { 1 }$ and a point $\mathrm { D }$ on circle $C _ { 2 }$ are given, with three points $\mathrm { A } , \mathrm { O } _ { 1 } , \mathrm { O } _ { 2 }$ and three points $\mathrm { C } , \mathrm { O } _ { 2 } , \mathrm { D }$ each on a line.
Let $\angle \mathrm { BO } _ { 1 } \mathrm {~A} = \theta _ { 1 } , \angle \mathrm { O } _ { 2 } \mathrm { O } _ { 1 } \mathrm { C } = \theta _ { 2 } , \angle \mathrm { O } _ { 1 } \mathrm { O } _ { 2 } \mathrm { D } = \theta _ { 3 }$.
The following is the process of finding the ratio of the lengths of segments AB and CD when $\overline { \mathrm { AB } } : \overline { \mathrm { O } _ { 1 } \mathrm { D } } = 1 : 2 \sqrt { 2 }$ and $\theta _ { 3 } = \theta _ { 1 } + \theta _ { 2 }$.
Since $\angle \mathrm { CO } _ { 2 } \mathrm { O } _ { 1 } + \angle \mathrm { O } _ { 1 } \mathrm { O } _ { 2 } \mathrm { D } = \pi$, we have $\theta _ { 3 } = \frac { \pi } { 2 } + \frac { \theta _ { 2 } } { 2 }$, and from $\theta _ { 3 } = \theta _ { 1 } + \theta _ { 2 }$, we get $2 \theta _ { 1 } + \theta _ { 2 } = \pi$, so $\angle \mathrm { CO } _ { 1 } \mathrm {~B} = \theta _ { 1 }$. Since $\angle \mathrm { O } _ { 2 } \mathrm { O } _ { 1 } \mathrm {~B} = \theta _ { 1 } + \theta _ { 2 } = \theta _ { 3 }$, triangles $\mathrm { O } _ { 1 } \mathrm { O } _ { 2 } \mathrm {~B}$ and $\mathrm { O } _ { 2 } \mathrm { O } _ { 1 } \mathrm { D }$ are congruent. Let $\overline { \mathrm { AB } } = k$. Since $\overline { \mathrm { BO } _ { 2 } } = \overline { \mathrm { O } _ { 1 } \mathrm { D } } = 2 \sqrt { 2 } k$, we have $\overline { \mathrm { AO } _ { 2 } } =$ (a) and since $\angle \mathrm { BO } _ { 2 } \mathrm {~A} = \frac { \theta _ { 1 } } { 2 }$, we have $\cos \frac { \theta _ { 1 } } { 2 } =$ (b). In triangle $\mathrm { O } _ { 2 } \mathrm { BC }$, with $\overline { \mathrm { BC } } = k , \overline { \mathrm { BO } _ { 2 } } = 2 \sqrt { 2 } k , \angle \mathrm { CO } _ { 2 } \mathrm {~B} = \frac { \theta _ { 1 } } { 2 }$, by the law of cosines, $\overline { \mathrm { O } _ { 2 } \mathrm { C } } =$ (c). Since $\overline { \mathrm { CD } } = \overline { \mathrm { O } _ { 2 } \mathrm { D } } + \overline { \mathrm { O } _ { 2 } \mathrm { C } } = \overline { \mathrm { O } _ { 1 } \mathrm { O } _ { 2 } } + \overline { \mathrm { O } _ { 2 } \mathrm { C } }$, $\overline { \mathrm { AB } } : \overline { \mathrm { CD } } = k : \left( \frac { \text{(a)} } { 2 } + \text{(c)} \right)$.
Let the expressions for (a) and (c) be $f ( k )$ and $g ( k )$ respectively, and let the number for (b) be $p$. What is the value of $f ( p ) \times g ( p )$? [4 points]
(1) $\frac { 169 } { 27 }$
(2) $\frac { 56 } { 9 }$
(3) $\frac { 167 } { 27 }$
(4) $\frac { 166 } { 27 }$
(5) $\frac { 55 } { 9 }$

For an ellipse $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 16 } = 1$ with foci $\mathrm { F } , \mathrm { F } ^ { \prime }$, there is a point A in the first quadrant on the ellipse. Among circles that are tangent to both lines $\mathrm { AF } , \mathrm { AF } ^ { \prime }$ and have their center on the y-axis, let C be the circle whose center has a negative y-coordinate. When the center of circle C is B and the area of quadrilateral $\mathrm { AFBF } ^ { \prime }$ is 72, what is the radius of circle C? [3 points]
(1) $\frac { 17 } { 2 }$
(2) 9
(3) $\frac { 19 } { 2 }$
(4) 10
(5) $\frac { 21 } { 2 }$

For the ellipse $\frac{x^2}{a^2} + \frac{y^2}{6} = 1$, what is the slope of the tangent line at the point $(\sqrt{3}, -2)$ on the ellipse? (where $a$ is a positive number) [3 points]
(1) $\sqrt{3}$
(2) $\frac{\sqrt{3}}{2}$
(3) $\frac{\sqrt{3}}{3}$
(4) $\frac{\sqrt{3}}{4}$
(5) $\frac{\sqrt{3}}{5}$

Let F be the focus of the parabola $y^2 = 8x$. From a point A on the parabola, drop a perpendicular to the directrix of the parabola, with the foot of the perpendicular being B. Let C and D be the two points where the line BF intersects the parabola. When $\overline{\mathrm{BC}} = \overline{\mathrm{CD}}$, what is the area of triangle ABD? (where $\overline{\mathrm{CF}} < \overline{\mathrm{DF}}$ and point A is not the origin) [3 points]
(1) $100\sqrt{2}$
(2) $104\sqrt{2}$
(3) $108\sqrt{2}$
(4) $112\sqrt{2}$
(5) $116\sqrt{2}$

As shown in the figure, there are two distinct planes $\alpha$ and $\beta$ with intersection line containing two points $\mathrm{A}$ and $\mathrm{B}$ where $\overline{\mathrm{AB}} = 18$. A circle $C_1$ with diameter AB lies on plane $\alpha$, and an ellipse $C_2$ with major axis AB and foci $\mathrm{F}$ and $\mathrm{F'}$ lies on plane $\beta$. Let H be the foot of the perpendicular from a point P on circle $C_1$ to plane $\beta$. Given that $\overline{\mathrm{HF'}} < \overline{\mathrm{HF}}$ and $\angle\mathrm{HFF'} = \frac{\pi}{6}$. Let Q be the point on ellipse $C_2$ where line HF intersects it, closer to H, with $\overline{\mathrm{FH}} < \overline{\mathrm{FQ}}$. The circle on plane $\beta$ centered at H passing through Q has radius 4 and is tangent to line AB. If the angle between the two planes $\alpha$ and $\beta$ is $\theta$, what is the value of $\cos\theta$? (where point P is not on plane $\beta$) [4 points]
(1) $\frac{2\sqrt{66}}{33}$
(2) $\frac{4\sqrt{69}}{69}$
(3) $\frac{\sqrt{2}}{3}$
(4) $\frac{4\sqrt{3}}{15}$
(5) $\frac{2\sqrt{78}}{39}$

For a natural number $n$ ($n \geq 2$), let the line $x = \frac{1}{n}$ meet the two ellipses $$C_{1} : \frac{x^{2}}{2} + y^{2} = 1, \quad C_{2} : 2x^{2} + \frac{y^{2}}{2} = 1$$ at points P and Q respectively in the first quadrant. Let $\alpha$ be the $x$-intercept of the tangent line to ellipse $C_{1}$ at point P, and let $\beta$ be the $x$-intercept of the tangent line to ellipse $C_{2}$ at point Q. How many values of $n$ satisfy $6 \leq \alpha - \beta \leq 15$? [3 points]
(1) 7
(2) 9
(3) 11
(4) 13
(5) 15

8. Circle C has its center on the line $2 x - y - 7 = 0$ and intersects the y-axis at two points $A ( 0 , - 4 )$ and $B ( 0 , - 2 )$. The equation of circle C is $\_\_\_\_$.

3. A moving point $P$ has equal distance to the point $F ( 2,0 )$ and to the line $x + 2 = 0$. Then the locus equation of $P$ is $y ^ { 2 } = 8 x$.
Analysis: This examines the definition and standard equation of a parabola. By the definition, the locus of $P$ is a parabola with focus $F ( 2,0 )$. Since $p = 2$, its equation is $y ^ { 2 } = 8 x$.

5. The distance from the center of circle $C : x ^ { 2 } + y ^ { 2 } - 2 x - 4 y + 4 = 0$ to the line $l : 3 x + 4 y + 4 = 0$ is $d =$ $\_\_\_\_$ $3$. Analysis: This examines the point-to-line distance formula. The distance from the center $(1,2)$ to the line $3 x + 4 y + 4 = 0$ is $\frac { | 3 \times 1 + 4 \times 2 + 4 | } { 5 } = 3$

7. The distance from the center of circle $C : x ^ { 2 } + y ^ { 2 } - 2 x - 4 y + 4 = 0$ to the line $3 x + 4 y + 4 = 0$ is $d =$ $\_\_\_\_$.