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]

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) $\frac { 25 } { 4 }$

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 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

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

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 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.

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

Given the hyperbola $C : \frac { x ^ { 2 } } { 3 } - y ^ { 2 } = 1$, with $O$ as the origin and $F$ as the right focus of $C$. A line through $F$ intersects the two asymptotes of $C$ at points $M$ and $N$. If $\triangle O M N$ is a right triangle, then $| M N | =$
A. $\frac { 3 } { 2 }$
B. 3
C. $2 \sqrt { 3 }$
D. 4

Let $A$ be a point on the parabola $C : y ^ { 2 } = 2 p x$ ($p > 0$). The distance from point $A$ to the focus of $C$ is 12, and the distance to the $y$-axis is 9. Then $p =$
A. 2
B. 3
C. 6
D. 9

Let $F$ be the focus of the parabola $C : y ^ { 2 } = 4 x$ , point $A$ is on $C$ , point $B ( 3,0 )$ , if $| A F | = | B F |$ , then $| A B | =$
A. 2
B. $2 \sqrt { 2 }$
C. 3
D. $3 \sqrt { 2 }$

Let $F$ be the focus of the parabola $C: y^2 = 4x$, point $A$ is on $C$, point $B(3,0)$. If $|AF| = |BF|$, then $|AB| =$
A. $2$
B. $2\sqrt{2}$
C. $3$
D. $3\sqrt{2}$

The parabola $y ^ { 2 } = 2 p x ( p > 0 )$ has its focus at a distance of $\sqrt { 2 }$ from the line $y = x + 1$. Then $p =$
A. 1
B. 2
C. $2 \sqrt { 2 }$
D. 4

The ellipse $\frac{x^{2}}{9} + \frac{y^{2}}{6} = 1$ has foci $F_{1} , F_{2}$ and center $O$ . Let $P$ be a point on the ellipse. If $\cos \angle F_{1}PF_{2} = \frac{3}{5}$ , then $|OP| =$
A. $\frac{2}{5}$
B. $\frac{\sqrt{30}}{2}$
C. $\frac{3}{5}$
D. $\frac{\sqrt{35}}{2}$

Given that point $A ( 1 , \sqrt { 5 } )$ lies on the parabola $C : y ^ { 2 } = 2 p x$, then the distance from $A$ to the directrix of $C$ is \_\_\_\_

Let the parabola $C: y^2 = 2px$ $(p > 0)$ have focus $F$. Point $A$ is on $C$. A perpendicular is drawn from $A$ to the directrix of $C$, with foot $B$. If the equation of line $BF$ is $y = -2x + 2$, then $|AF| = $ ( )
A. $3$
B. $4$
C. $5$
D. $6$

Let $e _ { 1 }$ and $e _ { 2 }$ be the eccentricities of the ellipse $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( b < 5 )$ and the hyperbola $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ respectively satisfying $\mathrm { e } _ { 1 } \mathrm { e } _ { 2 } = 1$. If $\alpha$ and $\beta$ are the distances between the foci of the ellipse and the foci of the hyperbola respectively, then the ordered pair $( \alpha , \beta )$ is equal to:
(1) $( 8,10 )$
(2) $\left( \frac { 20 } { 3 } , 12 \right)$
(3) $( 8,12 )$
(4) $\left( \frac { 24 } { 5 } , 10 \right)$

If the co-ordinates of two points $A$ and $B$ are $( \sqrt { 7 } , 0 )$ and $( - \sqrt { 7 } , 0 )$ respectively and $P$ is any point on the conic, $9 x ^ { 2 } + 16 y ^ { 2 } = 144$, then $PA + PB$ is equal to :
(1) 16
(2) 8
(3) 6
(4) 9

If the foci of a hyperbola are same as that of the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 25 } = 1$ and the eccentricity of the hyperbola is $\frac { 15 } { 8 }$ times the eccentricity of the ellipse, then the smaller focal distance of the point $\left(\sqrt { 2 } , \frac { 14 } { 3 } \sqrt { \frac { 2 } { 5 } }\right)$ on the hyperbola is equal to
(1) $7 \sqrt { \frac { 2 } { 5 } } - \frac { 8 } { 3 }$
(2) $14 \sqrt { \frac { 2 } { 5 } } - \frac { 4 } { 3 }$
(3) $14 \sqrt { \frac { 2 } { 5 } } - \frac { 16 } { 3 }$
(4) $7 \sqrt { \frac { 2 } { 5 } } + \frac { 8 } { 3 }$

Let $H : \frac { - x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ be the hyperbola, whose eccentricity is $\sqrt { 3 }$ and the length of the latus rectum is $4 \sqrt { 3 }$. Suppose the point $( \alpha , 6 ) , \alpha > 0$ lies on $H$. If $\beta$ is the product of the focal distances of the point $( \alpha , 6 )$, then $\alpha ^ { 2 } + \beta$ is equal to
(1) 172
(2) 171
(3) 169
(4) 170

The length of the latus rectum and directrices of a hyperbola with eccentricity e are 9 and $x = \pm \frac { 4 } { \sqrt { 13 } }$, respectively. Let the line $y - \sqrt { 3 } x + \sqrt { 3 } = 0$ touch this hyperbola at $( x _ { 0 } , y _ { 0 } )$. If m is the product of the focal distances of the point $\left( x _ { 0 } , y _ { 0 } \right)$, then $4 \mathrm { e } ^ { 2 } + \mathrm { m }$ is equal to $\_\_\_\_$