For the ellipse $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 20 } = 1$, let F and $\mathrm { F } ^ { \prime }$ be the two foci, and let A be the vertex closest to focus F. For a point P on this ellipse such that $\angle \mathrm { PFF } ^ { \prime } = \frac { \pi } { 3 }$, find the value of $\overline { \mathrm { PA } } ^ { 2 }$. [4 points]

Let $\mathrm { F } , \mathrm { F } ^ { \prime }$ be the two foci of the hyperbola $\frac { x ^ { 2 } } { 5 } - \frac { y ^ { 2 } } { 4 } = 1$, and let Q be the point symmetric to a point P on the hyperbola (not a vertex) with respect to the origin. When the area of quadrilateral $\mathrm { F } ^ { \prime } \mathrm { QFP }$ is 24, and the coordinates of point P are $( a , b )$, what is the value of $| a | + | b |$? [3 points]
(1) 9
(2) 10
(3) 11
(4) 12
(5) 13

The figure on the right shows 6 ellipses, each with a side of a regular hexagon ABCDEF with side length 10 as the major axis, and with equal minor axis lengths. As shown in the figure, the sum of the areas of 6 triangles formed by a vertex of the regular hexagon and the foci of the two adjacent ellipses is $6 \sqrt { 3 }$. What is the length of the minor axis of the ellipse? [3 points]
(1) $4 \sqrt { 2 }$
(2) 6
(3) $4 \sqrt { 3 }$
(4) 8
(5) $6 \sqrt { 2 }$

On a parabola $y ^ { 2 } = x$ with focus F, there is a point P such that $\overline { \mathrm { FP } } = 4$. As shown in the figure, point Q is taken on the extension of segment FP such that $\overline { \mathrm { FP } } = \overline { \mathrm { PQ } }$. What is the $x$-coordinate of point Q? [3 points]
(1) $\frac { 29 } { 4 }$
(2) 7
(3) $\frac { 27 } { 4 }$
(4) $\frac { 13 } { 2 }$
(5) $\frac { 25 } { 4 }$

The graph of the logarithmic function $y = \log _ { 2 } ( x + a ) + b$ passes through the focus of the parabola $y ^ { 2 } = x$, and the asymptote of the graph of this logarithmic function coincides with the directrix of the parabola $y ^ { 2 } = x$. What is the value of the sum $a + b$ of the two constants $a , b$? [3 points]
(1) $\frac { 5 } { 4 }$
(2) $\frac { 13 } { 8 }$
(3) $\frac { 9 } { 4 }$
(4) $\frac { 21 } { 8 }$
(5) $\frac { 11 } { 4 }$

An ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ is inscribed in a quadrilateral formed by connecting the four vertices of the ellipse $\frac { x ^ { 2 } } { 4 } + y ^ { 2 } = 1$. When the two foci of the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ are $\mathrm { F } ( b , 0 ) , \mathrm { F } ^ { \prime } ( - b , 0 )$, find the value of $a^2 + b^2$ (or the relevant quantity as stated in the problem). [3 points]

Let Q be the point where the tangent line at point $\mathrm { P } ( a , b )$ on the parabola $y ^ { 2 } = 4 x$ meets the $x$-axis. When $\overline { \mathrm { PQ } } = 4 \sqrt { 5 }$, what is the value of $a ^ { 2 } + b ^ { 2 }$? [3 points]
(1) 21
(2) 32
(3) 45
(4) 60
(5) 77

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) $\frac { 3 } { 4 } \pi$

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) $45 \sqrt { 2 }$

In coordinate space, there is a point $\mathrm { A } ( 9,0,5 )$, and on the $xy$-plane there is an ellipse $\frac { x ^ { 2 } } { 9 } + y ^ { 2 } = 1$. For a point P on the ellipse, find the maximum value of $\overline { \mathrm { AP } }$. [3 points]

Let $d$ be the distance between the focus of the parabola $y ^ { 2 } = n x$ and the tangent line to the parabola at the point $( n , n )$. Find the minimum natural number $n$ satisfying $d ^ { 2 } \geq 40$. [4 points]

The tangent line at the point $( b , 1 )$ on the hyperbola $x ^ { 2 } - 4 y ^ { 2 } = a$ is perpendicular to one asymptote of the hyperbola. What is the value of $a + b$? (Given that $a , b$ are positive numbers.) [3 points]
(1) 68
(2) 77
(3) 86
(4) 95
(5) 104

For a natural number $n$, a line passing through the focus F of the parabola $y ^ { 2 } = \frac { x } { n }$ intersects the parabola at two points P and Q, respectively. If $\overline { \mathrm { PF } } = 1$ and $\overline { \mathrm { FQ } } = a _ { n }$, what is the value of $\sum _ { n = 1 } ^ { 10 } \frac { 1 } { a _ { n } }$? [4 points]
(1) 210
(2) 205
(3) 200
(4) 195
(5) 190

For the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$, let F be the focus with positive $x$-coordinate and $\mathrm { F } ^ { \prime }$ be the focus with negative $x$-coordinate. A point P on this ellipse is chosen in the first quadrant such that $\angle \mathrm { FPF } ^ { \prime } = \frac { \pi } { 2 }$, and a point Q with positive $y$-coordinate is chosen on the extension of segment FP such that $\overline { \mathrm { FQ } } = 6$. Find the area of triangle $\mathrm { QF } ^ { \prime } \mathrm { F}$. [4 points]

As shown in the figure, there is an ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ with foci at $\mathrm { F } ( c , 0 )$ and $\mathrm { F } ^ { \prime } ( - c , 0 )$. For point P on the ellipse in the second quadrant, let Q be the midpoint of segment $\mathrm { PF } ^ { \prime }$, and let R be the point that divides segment PF internally in the ratio $1 : 3$. When $\angle \mathrm { PQR } = \frac { \pi } { 2 }$, $\overline { \mathrm { QR } } = \sqrt { 5 }$, and $\overline { \mathrm { RF } } = 9$, find the value of $a ^ { 2 } + b ^ { 2 }$. (Here, $a$, $b$, and $c$ are positive numbers.) [4 points]

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

A hyperbola has asymptotes with equations $y = \pm \frac { 4 } { 3 } x$ and two foci at $\mathrm { F } ( c , 0 )$, $\mathrm { F } ^ { \prime } ( - c , 0 )$ $(c > 0)$, and satisfies the following conditions.
(a) For a point P on the hyperbola, $\overline { \mathrm { PF } ^ { \prime } } = 30$ and $16 \leq \overline { \mathrm { PF } } \leq 20$.
(b) For the vertex A with positive $x$-coordinate, the length of segment AF is a natural number. Find the length of the major axis of this hyperbola. [4 points]

For the ellipse $\frac { ( x - 2 ) ^ { 2 } } { a } + \frac { ( y - 2 ) ^ { 2 } } { 4 } = 1$, the coordinates of the two foci are $( 6 , b ) , ( - 2 , b )$. What is the value of $ab$? (Here, $a$ is positive.) [3 points]
(1) 40
(2) 42
(3) 44
(4) 46
(5) 48

As shown in the figure, for a point P on the hyperbola $\frac { x ^ { 2 } } { 8 } - \frac { y ^ { 2 } } { 17 } = 1$ with foci $\mathrm { F } , \mathrm { F } ^ { \prime }$, there is a circle $C$ that is tangent to both line FP and line $\mathrm { F } ^ { \prime } \mathrm { P }$ simultaneously and has its center on the $y$-axis. For point Q, the point of tangency of line $\mathrm { F } ^ { \prime } \mathrm { P }$ with circle $C$, we have $\overline { \mathrm { F } ^ { \prime } \mathrm { Q } } = 5 \sqrt { 2 }$. Find the value of $\overline { \mathrm { FP } } ^ { 2 } + { \overline { \mathrm { F } ^ { \prime } \mathrm { P } } } ^ { 2 }$. (Here, $\overline { \mathrm { F } ^ { \prime } \mathrm { P } } < \overline { \mathrm { FP } }$) [4 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]

As shown in the figure, an ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { 25 } = 1$ has foci at $\mathrm { F } ( 0 , c ) , \mathrm { F } ^ { \prime } ( 0 , - c )$. Let A be the point with positive $x$-coordinate where the ellipse meets the $x$-axis. Let B be the intersection of the line $y = c$ and the line $\mathrm { AF } ^ { \prime}$, and let P be the point with positive $x$-coordinate where the line $y = c$ meets the ellipse. If the difference between the perimeter of triangle $\mathrm { BPF } ^ { \prime}$ and the perimeter of triangle BFA is 4, what is the area of triangle $\mathrm { AFF } ^ { \prime}$? (Given: $0 < a < 5 , c > 0$) [3 points]
(1) $3 \sqrt { 6 }$
(2) $\frac { 7 \sqrt { 6 } } { 2 }$
(3) $4 \sqrt { 6 }$
(4) $\frac { 9 \sqrt { 6 } } { 2 }$
(5) $5 \sqrt { 6 }$

In a plane, there is an equilateral triangle ABC with side length 10. For a point P satisfying $\overline { \mathrm { PB } } - \overline { \mathrm { PC } } = 2$, when the length of segment PA is minimized, what is the area of triangle PBC? [4 points]
(1) $20 \sqrt { 3 }$
(2) $21 \sqrt { 3 }$
(3) $22 \sqrt { 3 }$
(4) $23 \sqrt { 3 }$
(5) $24 \sqrt { 3 }$

For a hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { 6 } = 1$ with one focus at $( 3 \sqrt { 2 } , 0 )$, what is the length of the major axis? (Given that $a$ is a positive number.) [3 points]
(1) $3 \sqrt { 3 }$
(2) $\frac { 7 \sqrt { 3 } } { 2 }$
(3) $4 \sqrt { 3 }$
(4) $\frac { 9 \sqrt { 3 } } { 2 }$
(5) $5 \sqrt { 3 }$

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) $\frac { 21 } { 2 }$