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]

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

20. (This question is worth 13 points) The focus F of the parabola $\mathrm { C } _ { 1 } : \mathrm { X } ^ { 2 } = 4 \mathrm { y }$ is also a focus of the ellipse $\mathrm { C } _ { 2 } : \frac { y ^ { 2 } } { a ^ { 2 } } + \frac { X ^ { 2 } } { b ^ { 2 } } = 1 ( \mathrm { a } > \mathrm { b } > 0 )$. The common chord of $\mathrm { C } _ { 1 }$ and $\mathrm { C } _ { 2 }$ has length $2 \sqrt { 6 }$. A line $l$ through point F intersects $\mathrm { C } _ { 1 }$ at points $\mathrm { A } , \mathrm { B }$ and intersects $\mathrm { C } _ { 2 }$ at points $\mathrm { C } , \mathrm { D }$, with $\overrightarrow { B D }$ and $\overrightarrow { A C }$ in the same direction.
(1) Find the equation of $\mathrm { C } _ { 2 }$;
(2) If $| \mathrm { AC } | = | \mathrm { BD } |$, find the slope of line $l$.

15. A hyperbola passes through the point $( 4 , \sqrt { 3 } )$ and has asymptote equations $y = \pm \frac { 1 } { 2 } x$. The standard equation of this hyperbola is $\_\_\_\_$ .

20. (This question is worth 12 points). The ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ has eccentricity $\frac { \sqrt { 2 } } { 2 }$, and the point $( 2 , \sqrt { 2 } )$ lies

14. If the directrix of the parabola $y ^ { 2 } = 2 p x ( p > 0 )$ passes through a focus of the hyperbola $x ^ { 2 } - y ^ { 2 } = 1$, then $p = $ $\_\_\_\_$

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) $\frac { x ^ { 2 } } { 3 } - y ^ { 2 } = 1$

Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ $(a > 0, b > 0)$, one of its asymptotes passes through the point $(2, \sqrt{3})$, and one focus of the hyperbola lies on the directrix of the parabola $y^2 = 4\sqrt{7}x$. Then the equation of the hyperbola is
(A) $\frac{x^2}{21} - \frac{y^2}{28} = 1$
(B) $\frac{x^2}{28} - \frac{y^2}{21} = 1$
(C) $\frac{x^2}{3} - \frac{y^2}{4} = 1$
(D) $\frac{x^2}{4} - \frac{y^2}{3} = 1$

Let $O$ be the origin of coordinates. The line $x = 2$ intersects the parabola $C : y ^ { 2 } = 2 p x ( p > 0 )$ at points $D , E$. If $O D \perp O E$, then the focus coordinates of $C$ are
A. $\left( \frac { 1 } { 4 } , 0 \right)$
B. $\left( \frac { 1 } { 2 } , 0 \right)$
C. $( 1,0 )$
D. $( 2,0 )$

Let $O$ be the origin of coordinates. The line $x = 2$ intersects the parabola $C : y ^ { 2 } = 2 p x ( p > 0 )$ at points $D$ and $E$ . If $O D \perp O E$ , then the focus coordinates of $C$ are
A. $\left( \frac { 1 } { 4 } , 0 \right)$
B. $\left( \frac { 1 } { 2 } , 0 \right)$
C. $( 1,0 )$
D. $( 2,0 )$

The ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ has eccentricity $\frac { 1 } { 3 }$. Let $A _ { 1 } , A _ { 2 }$ be the left and right vertices of $C$ respectively, and $B$ be the upper vertex. If $\overrightarrow { B A _ { 1 } } \cdot \overrightarrow { B A _ { 2 } } = - 1$ , then the equation of $C$ is
A. $\frac { x ^ { 2 } } { 18 } + \frac { y ^ { 2 } } { 16 } = 1$
B. $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 8 } = 1$
C. $\frac { x ^ { 2 } } { 3 } + \frac { y ^ { 2 } } { 2 } = 1$
D. $\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 32 } = 1$

An ellipse $E$ has its center at the origin, with axes of symmetry along the $x$-axis and $y$-axis, and passes through points $A ( 0 , - 2 ) , B \left( \frac { 3 } { 2 } , 1 \right)$.
(The remainder of this question was cut off in the source document.)

Given the ellipse $C : \frac { y ^ { 2 } } { a ^ { 2 } } + \frac { x ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ with eccentricity $\frac { \sqrt { 5 } } { 3 }$, and point $A ( - 2,0 )$ lies on $C$.
(1) Find the equation of $C$.
(2) A line passing through point $( - 2,3 )$ intersects $C$ at points $P$ and $Q$. Lines $AP$ and $AQ$ intersect the $y$-axis at points $M$ and $N$ respectively. Prove that the midpoint of segment $MN$ is a fixed point.

137- The two lines $y = -2x$ and $y = 2x + 4$ are the asymptotes of a hyperbola, and $M\!\left(\dfrac{3}{2},\, 5\right)$ is one of its points. The distance between the two foci of this hyperbola is:
(1) $2\sqrt{3}$ (2) $2\sqrt{5}$ (3) $4\sqrt{3}$ (4) $4\sqrt{5}$
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137- For which value of $a$, the distance between the foci of the hyperbola $3x^2 + 4y^2 + 16y + a = 0$ equals $2$?
(1) $2$ (2) $4$ (3) $6$ (4) $8$

134- The parabola with focus $F(2,1)$ and directrix $x = 4$, the equation is which of the following?
(1) $y^2 - 2y + 4x = 11$ (2) $y^2 - 2y + 3x = 5$
(3) $x^2 - 4x + 4y = \circ$ (4) $x^2 - 6x + 2y = -4$

136- If the point $F(-0.25\ ,\ -2)$ is the focus of the parabola $y^2 + ay + bx + 1 = 0$, what is the smallest value of $b$?

• [(1)] $-4$
• [(2)] $-3$
• [(3)] $-2$
• [(4)] $2$

146. The parabola $6 = 6y - 12y - (x-1)^2$ has vertex $F$ and focus $F'$. An ellipse has foci $F$ and $F'$ and eccentricity $0.6$. What is the distance from the center of the ellipse to the origin?
(1) $1$ (2) $\sqrt{2}$ (3) $\sqrt{3}$ (4) $2$

7. On July 5 next, the Earth will reach aphelion, the point of its orbit where the distance from the Sun is maximum, approximately $1.52 \cdot 10 ^ { 11 } \mathrm {~m}$. Perihelion is instead the point at minimum distance from the Sun, approximately $1.47 \cdot 10 ^ { 11 } \mathrm {~m}$. Determine, in an appropriate coordinate system, the equation that represents the Earth's trajectory around the Sun.

An ellipse intersects the hyperbola $2x^{2}-2y^{2}=1$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are along the coordinate axes, then
(A) Equation of ellipse is $x^{2}+2y^{2}=2$
(B) The foci of ellipse are $(\pm1,0)$
(C) Equation of ellipse is $x^{2}+2y^{2}=4$
(D) The foci of ellipse are $(\pm\sqrt{2},0)$

Equation of the ellipse whose axes are the axes of coordinates and which passes through the point $(-3,1)$ and has eccentricity $\sqrt{\frac{2}{5}}$ is
(1) $5x^{2}+3y^{2}-48=0$
(2) $3x^{2}+5y^{2}-15=0$
(3) $5x^{2}+3y^{2}-32=0$
(4) $3x^{2}+5y^{2}-32=0$

If the eccentricity of a hyperbola $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$, which passes through $( K , 2 )$, is $\frac { \sqrt { 13 } } { 3 }$, then the value of $K ^ { 2 }$ is
(1) 18
(2) 8
(3) 1
(4) 2