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) $6 \sqrt { 2 }$

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) $45 \sqrt { 2 }$

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, 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) $5 \sqrt { 6 }$

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) $116\sqrt{2}$

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]

Given that $F$ is the right focus of the hyperbola $C: x^2 - \frac{y^2}{3} = 1$, $P$ is a point on $C$, and $PF$ is perpendicular to the $x$-axis. Point $A$ has coordinates $(1, 3)$. Then the area of $\triangle APF$ is
A. $\frac{3}{2}$
B. $\frac{1}{2}$
C. $\frac{2}{3}$
D. $\frac{3}{4}$

The right focus of the hyperbola $C : \frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 2 } = 1$ is $F$. Point $P$ is on one of the asymptotes of $C$, and $O$ is the origin. If $| PO | = | PF |$, then the area of $\triangle PFO$ is
A. $\frac { 3 \sqrt { 2 } } { 4 }$
B. $\frac { 3 \sqrt { 2 } } { 2 }$
C. $2 \sqrt { 2 }$
D. $3 \sqrt { 2 }$

Let $F _ { 1 } , F _ { 2 }$ be the two foci of the ellipse $C : \frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 20 } = 1$ , and $M$ be a point on $C$ in the first quadrant. If $\triangle M F _ { 1 } F _ { 2 }$ is an isosceles triangle, then the coordinates of $M$ are \_\_\_\_\_\_.

10. Let $F$ be a focus of the hyperbola $C : \frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 5 } = 1$ . Point $P$ is on $C$ , $O$ is the origin. If $| O P | = | O F |$ , then the area of $\triangle O P F$ is
A. $\frac { 3 } { 2 }$
B. $\frac { 5 } { 2 }$
C. $\frac { 7 } { 2 }$
D. $\frac { 9 } { 2 }$

10. For the hyperbola $C : \frac { x ^ { 2 } } { 4 } - \frac { y ^ { 2 } } { 2 } = 1$ with right focus $F$ , if point $P$ is on one of the asymptotes of $C$ , $O$ is the origin, and $| P O | = | P F |$ , then the area of $\triangle P F O$ is
A. $\frac { 3 \sqrt { 2 } } { 4 }$
B. $\frac { 3 \sqrt { 2 } } { 2 }$
C. $2 \sqrt { 2 }$
D. $3 \sqrt { 2 }$

Let $F _ { 1 } , F _ { 2 }$ be the two foci of the hyperbola $C : x ^ { 2 } - \frac { y ^ { 2 } } { 3 } = 1$ , $O$ be the origin, and point $P$ on $C$ with $| O P | = 2$ . The area of $\triangle P F _ { 1 } F _ { 2 }$ is
A. $\frac { 7 } { 2 }$
B. 3
C. $\frac { 5 } { 2 }$
D. 2

Given the ellipse $C : \frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { m ^ { 2 } } = 1 ( 0 < m < 5 )$ with eccentricity $\frac { \sqrt { 15 } } { 4 }$, where $A , B$ are the left and right vertices of $C$ respectively.
(1) Find the equation of $C$;
(2) If point $P$ is on $C$, point $Q$ is on the line $x = 6$, and $| B P | = | B Q | , B P \perp B Q$, find the area of $\triangle A P Q$ .

(15 points) Given that $A ( 0,3 )$ and $P \left( 3 , \frac { 3 } { 2 } \right)$ are two points on the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ .
(1) Find the eccentricity of $C$ ;
(2) If a line $l$ through $P$ intersects $C$ at another point $B$ , and the area of $\triangle A B P$ is 9 , find the equation of $l$ .

Consider a branch of the hyperbola
$$x ^ { 2 } - 2 y ^ { 2 } - 2 \sqrt { 2 } x - 4 \sqrt { 2 } y - 6 = 0$$
with vertex at the point $A$. Let $B$ be one of the end points of its latus rectum. If $C$ is the focus of the hyperbola nearest to the point $A$, then the area of the triangle $A B C$ is
(A) $1 - \sqrt { \frac { 2 } { 3 } }$
(B) $\sqrt { \frac { 3 } { 2 } } - 1$
(C) $1 + \sqrt { \frac { 2 } { 3 } }$
(D) $\sqrt { \frac { 3 } { 2 } } + 1$

The line passing through the extremity $A$ of the major axis and extremity $B$ of the minor axis of the ellipse
$$x ^ { 2 } + 9 y ^ { 2 } = 9$$
meets its auxiliary circle at the point $M$. Then the area of the triangle with vertices at $A , M$ and the origin $O$ is
(A) $\frac { 31 } { 10 }$
(B) $\frac { 29 } { 10 }$
(C) $\frac { 21 } { 10 }$
(D) $\frac { 27 } { 10 }$

A vertical line passing through the point $( h , 0 )$ intersects the ellipse $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 1$ at the points $P$ and $Q$. Let the tangents to the ellipse at $P$ and $Q$ meet at the point $R$. If $\Delta ( h ) =$ area of the triangle $P Q R$, $\Delta _ { 1 } = \max _ { 1/2 \leq h \leq 1 } \Delta ( h )$ and $\Delta _ { 2 } = \min _ { 1/2 \leq h \leq 1 } \Delta ( h )$, then $\frac { 8 } { \sqrt { 5 } } \Delta _ { 1 } - 8 \Delta _ { 2 } =$

Let $F _ { 1 } \left( x _ { 1 } , 0 \right)$ and $F _ { 2 } \left( x _ { 2 } , 0 \right)$, for $x _ { 1 } < 0$ and $x _ { 2 } > 0$, be the foci of the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 8 } = 1$. Suppose a parabola having vertex at the origin and focus at $F _ { 2 }$ intersects the ellipse at point $M$ in the first quadrant and at point $N$ in the fourth quadrant.
The orthocentre of the triangle $F _ { 1 } M N$ is
(A) $\left( - \frac { 9 } { 10 } , 0 \right)$
(B) $\left( \frac { 2 } { 3 } , 0 \right)$
(C) $\left( \frac { 9 } { 10 } , 0 \right)$
(D) $\left( \frac { 2 } { 3 } , \sqrt { 6 } \right)$

Let $F _ { 1 } \left( x _ { 1 } , 0 \right)$ and $F _ { 2 } \left( x _ { 2 } , 0 \right)$, for $x _ { 1 } < 0$ and $x _ { 2 } > 0$, be the foci of the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 8 } = 1$. Suppose a parabola having vertex at the origin and focus at $F _ { 2 }$ intersects the ellipse at point $M$ in the first quadrant and at point $N$ in the fourth quadrant.
If the tangents to the ellipse at $M$ and $N$ meet at $R$ and the normal to the parabola at $M$ meets the $x$-axis at $Q$, then the ratio of area of the triangle $M Q R$ to area of the quadrilateral $M F _ { 1 } N F _ { 2 }$ is
(A) $3 : 4$
(B) $4 : 5$
(C) $5 : 8$
(D) $2 : 3$

Consider the hyperbola
$$\frac { x ^ { 2 } } { 100 } - \frac { y ^ { 2 } } { 64 } = 1$$
with foci at $S$ and $S _ { 1 }$, where $S$ lies on the positive $x$-axis. Let $P$ be a point on the hyperbola, in the first quadrant. Let $\angle S P S _ { 1 } = \alpha$, with $\alpha < \frac { \pi } { 2 }$. The straight line passing through the point $S$ and having the same slope as that of the tangent at $P$ to the hyperbola, intersects the straight line $S _ { 1 } P$ at $P _ { 1 }$. Let $\delta$ be the distance of $P$ from the straight line $S P _ { 1 }$, and $\beta = S _ { 1 } P$. Then the greatest integer less than or equal to $\frac { \beta \delta } { 9 } \sin \frac { \alpha } { 2 }$ is $\_\_\_\_$ .

The area of triangle formed by the lines joining the vertex of the parabola, $x^{2} = 8y$, to the extremities of its latus rectum is
(1) 2
(2) 8
(3) 1
(4) 4

Tangents are drawn to the hyperbola $4 x ^ { 2 } - y ^ { 2 } = 36$ at the points $P$ and $Q$. If these tangents intersect at the point $T ( 0,3 )$ then the area (in sq. units) of $\triangle P T Q$ is:
(1) $36 \sqrt { 5 }$
(2) $45 \sqrt { 5 }$
(3) $54 \sqrt { 3 }$
(4) $60 \sqrt { 3 }$

Consider a hyperbola $H : x ^ { 2 } - 2 y ^ { 2 } = 4$. Let the tangent at a point $P ( 4 , \sqrt { 6 } )$ meet the $x$-axis at $Q$ and latus rectum at $R \left( x _ { 1 } , y _ { 1 } \right) , x _ { 1 } > 0$. If $F$ is a focus of $H$ which is nearer to the point $P$, then the area of $\triangle QFR$ (in sq. units) is equal to
(1) $4 \sqrt { 6 }$
(2) $\sqrt { 6 } - 1$
(3) $\frac { 7 } { \sqrt { 6 } } - 2$
(4) $4 \sqrt { 6 } - 1$

Let the tangent to the parabola $S : y ^ { 2 } = 2 x$ at the point $P ( 2,2 )$ meet the $x$-axis at $Q$ and normal at it meet the parabola $S$ at the point $R$. Then the area (in sq. units) of the triangle $P Q R$ is equal to:
(1) $\frac { 25 } { 2 }$
(2) $\frac { 35 } { 2 }$
(3) $\frac { 15 } { 2 }$
(4) 25