Let the foci of a hyperbola be $( 1,14 )$ and $( 1 , - 12 )$. If it passes through the point $( 1,6 )$, then the length of its latus-rectum is:
(1) $\frac { 24 } { 5 }$
(2) $\frac { 25 } { 6 }$
(3) $\frac { 144 } { 5 }$
(4) $\frac { 288 } { 5 }$

If $A$ and $B$ are the points of intersection of the circle $x ^ { 2 } + y ^ { 2 } - 8 x = 0$ and the hyperbola $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$ and a point P moves on the line $2 x - 3 y + 4 = 0$, then the centroid of $\triangle \mathrm { PAB }$ lies on the line :
(1) $x + 9 y = 36$
(2) $4 x - 9 y = 12$
(3) $6 x - 9 y = 20$
(4) $9 x - 9 y = 32$

If the equation of the parabola with vertex $\mathrm{V}\left(\frac{3}{2}, 3\right)$ and the directrix $x + 2y = 0$ is $\alpha x^{2} + \beta y^{2} - \gamma xy - 30x - 60y + 225 = 0$, then $\alpha + \beta + \gamma$ is equal to:
(1) 7
(2) 9
(3) 8
(4) 6

Let $\mathrm { E } : \frac { x ^ { 2 } } { \mathrm { a } ^ { 2 } } + \frac { y ^ { 2 } } { \mathrm {~b} ^ { 2 } } = 1 , \mathrm { a } > \mathrm { b }$ and $\mathrm { H } : \frac { x ^ { 2 } } { \mathrm {~A} ^ { 2 } } - \frac { y ^ { 2 } } { \mathrm {~B} ^ { 2 } } = 1$. Let the distance between the foci of E and the foci of $H$ be $2 \sqrt { 3 }$. If $a - A = 2$, and the ratio of the eccentricities of $E$ and $H$ is $\frac { 1 } { 3 }$, then the sum of the lengths of their latus rectums is equal to:
(1) 10
(2) 9
(3) 8
(4) 7

Let A and B be the two points of intersection of the line $y + 5 = 0$ and the mirror image of the parabola $y ^ { 2 } = 4 x$ with respect to the line $x + y + 4 = 0$. If d denotes the distance between A and B, and a denotes the area of $\triangle S A B$, where $S$ is the focus of the parabola $y ^ { 2 } = 4 x$, then the value of $( a + d )$ is

Let $\mathrm{H}_{1} : \frac{x^{2}}{\mathrm{a}^{2}} - \frac{y^{2}}{\mathrm{b}^{2}} = 1$ and $\mathrm{H}_{2} : -\frac{x^{2}}{\mathrm{A}^{2}} + \frac{y^{2}}{\mathrm{B}^{2}} = 1$ be two hyperbolas having length of latus rectums $15\sqrt{2}$ and $12\sqrt{5}$ respectively. Let their eccentricities be $e_{1} = \sqrt{\frac{5}{2}}$ and $e_{2}$ respectively. If the product of the lengths of their transverse axes is $100\sqrt{10}$, then $25\mathrm{e}_{2}^{2}$ is equal to $\_\_\_\_$.

Consider the following two equations
$$\begin{gathered} \left( \log _ { 4 } 2 \sqrt { x } \right) ^ { 2 } + \left( \log _ { 4 } 2 \sqrt { y } \right) ^ { 2 } = \log _ { 2 } ( \sqrt [ 4 ] { 2 } \cdot x \sqrt { y } ) \\ \sqrt [ 3 ] { x } \cdot \sqrt [ 4 ] { y } = 2 ^ { k } \end{gathered}$$
We are to find the range of values which the constant $k$ can take so that there exist positive real numbers $x , y$ which satisfy (1) and (2) simultaneously.
Set $X = \log _ { 2 } x$ and $Y = \log _ { 2 } y$. Let us express (1) and (2) in terms of $X$ and $Y$. First we consider (1). Since
$$\log _ { 4 } 2 \sqrt { x } = \frac { \log _ { 2 } x + \mathbf { A } } { \mathbf { B } }$$
and
$$\log _ { 2 } ( \sqrt [ 4 ] { 2 } \cdot x \sqrt { y } ) = \frac { \mathbf { C } } { \mathbf { D } } + \log _ { 2 } x + \frac { \log _ { 2 } y } { \mathbf { E } } ,$$
(1) reduces to
$$( X - \mathbf { F } ) ^ { 2 } + ( Y - \mathbf { G } ) ^ { 2 } = \mathbf { H I } .$$
In the same way, (2) reduces to
$$4 X + \mathbf { J } Y = \mathbf { K } \mathbf { L } k .$$
Since the distance $d$ from the center of the circle (3) to the straight line (4) on the $XY$-plane is given by
$$d = \frac { | \mathbf { M N } - \mathbf { O P } k | } { \mathbf { Q } } ,$$
the range of values which $k$ can take is
$$\mathbf { R } \leq k \leqq \mathbf { S } .$$

Suppose $A, B$ are two points on a parabola $\Gamma$ and their connecting line segment passes through the focus $F$ of $\Gamma$. Let the projections of $A, F, B$ onto the directrix of $\Gamma$ be $A', F', B'$ respectively. Select the option equal to $\frac{\overline{AF}}{\overline{A'F'}}$. (Note: This schematic diagram only illustrates the relative positions of the points; the distance relationships between points are not accurate)
(1) $\tan \angle 1$, where $\angle 1 = \angle A'F'A$
(2) $\sin \angle 2$, where $\angle 2 = \angle AF'F$
(3) $\sin \angle 3$, where $\angle 3 = \angle A'AF$
(4) $\cos \angle 4$, where $\angle 4 = \angle F'FB$
(5) $\tan \angle 5$, where $\angle 5 = \angle FF'B$

A flashlight's light beam forms a right circular cone with a light divergence angle of $60^{\circ}$, as shown in the figure. The wall is perpendicular to the floor, and their intersection is a straight line $L$. The flashlight is directed perpendicular to $L$, meaning the axis of the right circular cone is perpendicular to $L$. If the edge of the light beam on the wall is part of a parabola, then the edge of the light beam on the floor is part of which of the following shapes?
(1) Two intersecting lines (2) Circle (3) Parabola (4) Ellipse with unequal major and minor axes (5) Hyperbola

On the coordinate plane, let $\Gamma$ be an ellipse with center at the origin and major axis on the $y$-axis. It is known that a linear transformation of counterclockwise rotation by angle $\theta$ about the origin (where $0 < \theta < \pi$) transforms $\Gamma$ to a new ellipse $\Gamma ^ { \prime } : 40 x ^ { 2 } + 4 \sqrt { 5 } x y + 41 y ^ { 2 } = 180$. The point $\left( - \frac { 5 } { 3 } , \frac { 2 \sqrt { 5 } } { 3 } \right)$ is one of the two points on $\Gamma ^ { \prime }$ farthest from the origin.
The length of the major axis of ellipse $\Gamma ^ { \prime }$ is (15-1) $\sqrt{\underline{(15-2)}}$. (Express as a simplified radical)

On the coordinate plane, let $\Gamma$ be an ellipse with center at the origin and major axis on the $y$-axis. It is known that a linear transformation of counterclockwise rotation by angle $\theta$ about the origin (where $0 < \theta < \pi$) transforms $\Gamma$ to a new ellipse $\Gamma ^ { \prime } : 40 x ^ { 2 } + 4 \sqrt { 5 } x y + 41 y ^ { 2 } = 180$. The point $\left( - \frac { 5 } { 3 } , \frac { 2 \sqrt { 5 } } { 3 } \right)$ is one of the two points on $\Gamma ^ { \prime }$ farthest from the origin.
Find the equation of the line containing the minor axis of $\Gamma ^ { \prime }$ and the length of the minor axis.

On the coordinate plane, the equation of ellipse $\Gamma$ is $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{6^{2}} = 1$ (where $a$ is a positive real number). If $\Gamma$ is scaled by a factor of 2 in the $x$-axis direction and by a factor of 3 in the $y$-axis direction with the origin $O$ as the center, the resulting new figure passes through the point $(18, 0)$. Which of the following options is a focus of $\Gamma$?
(1) $(0, 3\sqrt{3})$
(2) $(-3\sqrt{5}, 0)$
(3) $(0, 6\sqrt{13})$
(4) $(-3\sqrt{13}, 0)$
(5) $(9, 0)$

From the following conic sections on the coordinate plane, select the options that intersect all vertical lines.
(1) $\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$
(2) $\frac{x^{2}}{9} - \frac{y^{2}}{4} = 1$
(3) $-\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1$
(4) $y = \frac{4}{9}x^{2}$
(5) $x = \frac{4}{9}y^{2}$

In space, two intersecting lines $L , M$ form an angle of $24 ^ { \circ }$ . Rotating $M$ around $L$ one complete revolution generates a right circular cone surface. A plane $E$ is parallel to line $L$. What is the cross-section formed by plane $E$ and this cone surface?
(1) Hyperbola
(2) Parabola
(3) Ellipse (with unequal major and minor axes)
(4) Circle
(5) Two intersecting lines

Answer the following questions regarding curves on the $x y$-plane.
(1) Show that the foci of an ellipse:
$$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 \quad ( a > b > 0 )$$
and those of a hyperbola:
$$\frac { x ^ { 2 } } { c ^ { 2 } } - \frac { y ^ { 2 } } { d ^ { 2 } } = 1 \quad ( c > d > 0 )$$
are $\left( \pm \sqrt { a ^ { 2 } - b ^ { 2 } } , 0 \right)$ and $\left( \pm \sqrt { c ^ { 2 } + d ^ { 2 } } , 0 \right)$, respectively. Note that an ellipse (hyperbola) is a curve such that the sum (difference) of the distances from the foci to any point on the curve is constant.
(2) As for the ellipse equation, consider the set $E _ { u }$ of ellipses such that $a ^ { 2 } - b ^ { 2 } = u ^ { 2 }$ ($u$ is a positive constant). By writing the simultaneous equations that consist of the ellipse equation and the differential equation obtained by taking the derivative of the ellipse equation with respect to $x$, show that any ellipse in $E _ { u }$ satisfies
$$x y y ^ { \prime 2 } + \left( x ^ { 2 } - y ^ { 2 } - u ^ { 2 } \right) y ^ { \prime } - x y = 0 , \quad ( * * * )$$
where $y ^ { \prime } = \frac { \mathrm { d} y } { \mathrm {~d} x }$.
(3) As for the hyperbola equation, consider the set $H _ { u }$ of hyperbolae such that $c ^ { 2 } + d ^ { 2 } = u ^ { 2 }$. Show that any hyperbola in $H _ { u }$ satisfies Eq. $(***) $.
(4) Let $C _ { u }$ be the set of curves perpendicular to any ellipse in $E _ { u }$. Let $D _ { u }$ be the set of curves obtained by removing from $C _ { u }$ the line $x = 0$ as well as all the curves including a point such that $y ^ { \prime } = 0$. Find a differential equation that any curve in $D _ { u }$ satisfies.
(5) Solve the differential equation that you found in Question (4). If necessary, rewrite the differential equation into a differential equation with respect to $p$ with replacement such that $\alpha = x ^ { 2 } , \beta = y ^ { 2 }$, and $p = \frac { \mathrm { d} \beta } { \mathrm { d} \alpha }$.

In the Cartesian coordinate plane, an ellipse with center at the origin and foci at points E and F is given below. The vertical line drawn from point F intersects the ellipse at points, and the point with positive y-coordinate is denoted by K. The equation of the line passing through points K and E is $\mathrm { y } = \mathrm { x } + 1$.
Accordingly, what is the value of a?
A) $\sqrt { 2 } + 1$ B) $\sqrt { 3 } + 2$ C) $\sqrt { 5 } + 1$ D) $3 - \sqrt { 2 }$