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) $6 \sqrt { 2 }$

Let $\mathrm { F } , \mathrm { F } ^ { \prime }$ be the two foci of the ellipse $\frac { x ^ { 2 } } { 4 } + y ^ { 2 } = 1$. For a point P on this ellipse satisfying $| \overrightarrow { \mathrm { OP } } + \overrightarrow { \mathrm { OF } } | = 1$, the length of segment PF is $k$. Find the value of $5k$. (Here, O is the origin.)

As shown in the figure, let $\mathrm { F } , \mathrm { F } ^ { \prime }$ be the two foci of the hyperbola $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$. For point P on the hyperbola in the first quadrant and point Q on the hyperbola in the second quadrant, when $\overline { \mathrm { PF } ^ { \prime } } - \overline { \mathrm { QF } ^ { \prime } } = 3$, find the value of $\overline { \mathrm { QF } } - \overline { \mathrm { PF } }$. [3 points]

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 $d$ 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

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

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]

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]

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 two positive numbers $a , p$, let $\mathrm { F } _ { 1 }$ be the focus of the parabola $( y - a ) ^ { 2 } = 4 p x$, and let $\mathrm { F } _ { 2 }$ be the focus of the parabola $y ^ { 2 } = - 4 x$.
When segment $\mathrm { F } _ { 1 } \mathrm {~F} _ { 2 }$ meets the two parabolas at points $\mathrm { P } , \mathrm { Q }$ respectively, $\overline { \mathrm { F } _ { 1 } \mathrm {~F} _ { 2 } } = 3$ and $\overline { \mathrm { PQ } } = 1$. What is the value of $a ^ { 2 } + p ^ { 2 }$? [4 points]
(1) 6
(2) $\frac { 25 } { 4 }$
(3) $\frac { 13 } { 2 }$
(4) $\frac { 27 } { 4 }$
(5) 7

For a positive number $c$, there is a hyperbola with foci $\mathrm{F}(c, 0)$ and $\mathrm{F'}(-c, 0)$ and major axis length 6. Two distinct points $\mathrm{P}$ and $\mathrm{Q}$ on this hyperbola satisfy the following conditions. Find the sum of all values of $c$. [4 points] (가) Point P is in the first quadrant, and point Q is on line $\mathrm{PF'}$. (나) Triangle $\mathrm{PF'F}$ is isosceles. (다) The perimeter of triangle PQF is 28.

There is a hyperbola $x^{2} - \frac{y^{2}}{35} = 1$ with foci at $\mathrm{F}(c, 0)$, $\mathrm{F}'(-c, 0)$ ($c > 0$). For a point P on this hyperbola in the first quadrant, let Q be a point on line $\mathrm{PF}'$ such that $\overline{\mathrm{PQ}} = \overline{\mathrm{PF}}$. When triangle $\mathrm{QF'F}$ and triangle $\mathrm{FF'P}$ are similar, the area of triangle PFQ is $\frac{q}{p}\sqrt{5}$. Find the value of $p + q$. (Here, $\overline{\mathrm{PF}'} < \overline{\mathrm{QF}'}$ and $p$ and $q$ are coprime natural numbers.) [4 points]

13. In the Cartesian coordinate plane, the hyperbola $\Gamma$ is centered at the origin with one focus at $( \sqrt { 5 } , 0 )$ . $\overrightarrow { e _ { 1 } } = ( 2,1 )$ and $\overrightarrow { e _ { 2 } } = ( 2 , - 1 )$ are direction vectors of the two asymptotes respectively. For any point $P$ on the hyperbola $\Gamma$ , if $\overrightarrow { O P } = a \overrightarrow { e _ { 1 } } + b \overrightarrow { e _ { 2 } } ( a , b \in \mathbf { R } )$ , then an equation satisfied by $a$ and $b$ is $\_\_\_\_$.

13. As shown in the figure, the line $x = 2$ intersects the asymptotes of the hyperbola $\Gamma : \frac { x ^ { 2 } } { 4 } - y ^ { 2 } = 1$ at points $E _ { 1 }$ and $E _ { 2 }$. Let $\overrightarrow { O E _ { 1 } } = \overrightarrow { e _ { 1 } }$ and $\overrightarrow { O E _ { 2 } } = \overrightarrow { e _ { 2 } }$. For any point $P$ on the hyperbola $\Gamma$, if $\overrightarrow { O P } = a \overrightarrow { e _ { 1 } } + b \overrightarrow { e _ { 2 } }$ ($a , b \in \mathbb{R}$), then $a$ and $b$ satisfy the equation $\_\_\_\_$ $4 a b = 1$.
Analysis: $E _ { 1 } ( 2,1 )$, $E _ { 2 } ( 2 , - 1 )$
$$\overrightarrow { O P } = a \overrightarrow { e _ { 1 } } + b \overrightarrow { e _ { 2 } } = ( 2 a + 2 b , a - b ), \text{ point } P \text{ is on the hyperbola}$$
$\therefore \frac { ( 2 a + 2 b ) ^ { 2 } } { 4 } - ( a - b ) ^ { 2 } = 1$, which simplifies to $4 a b = 1$ [Figure]

3. If the hyperbola $E : \frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$ has left and right foci $F _ { 1 }$ and $F _ { 2 }$ respectively, point $P$ is on the hyperbola $E$, and $\left| P F _ { 1 } \right| = 3$, then $\left| P F _ { 2 } \right|$ equals
A. 11
B. 9
C. 5
D. 3

5. A line passing through the right focus of the hyperbola $x ^ { 2 } - \frac { y ^ { 2 } } { 3 } = 1$ and perpendicular to the $x$-axis intersects the two asymptotes of the hyperbola at points $\mathrm { A }$ and $\mathrm { B }$, then $| A B | =$
(A) $\frac { 4 \sqrt { 3 } } { 3 }$
(B) $2 \sqrt { 3 }$
(C) $6$
(D) $4 \sqrt { 3 }$

5. Given the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > 0 , b > 0 )$ with one focus at $F ( 2,0 )$, and the asymptote of the hyperbola is tangent to the circle $( x - 2 ) ^ { 2 } + y ^ { 2 } = 3$, then the equation of the hyperbola is
(A) $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 13 } = 1$
(B) $\frac { x ^ { 2 } } { 13 } - \frac { y ^ { 2 } } { 9 } = 1$
(C) $\frac { x ^ { 2 } } { 3 } - y ^ { 2 } = 1$