For a natural number $n$, a line passing through the focus F of the parabola $y ^ { 2 } = \frac { x } { n }$ intersects the parabola at two points P and Q, respectively. If $\overline { \mathrm { PF } } = 1$ and $\overline { \mathrm { FQ } } = a _ { n }$, what is the value of $\sum _ { n = 1 } ^ { 10 } \frac { 1 } { a _ { n } }$? [4 points]
(1) 210
(2) 205
(3) 200
(4) 195
(5) 190

Let $F$ be the focus of the parabola $C : y ^ { 2 } = 4 x$. Two perpendicular lines $l _ { 1 }$ and $l _ { 2 }$ pass through $F$. Line $l _ { 1 }$ intersects $C$ at points $A$ and $B$, and line $l _ { 2 }$ intersects $C$ at points $D$ and $E$. The minimum value of $|AB| + |DE|$ is
A. 16
B. 14
C. 12
D. 10

Given point $M ( - 1, 1 )$ and parabola $C : y ^ { 2 } = 4 x$. A line through the focus of $C$ with slope $k$ intersects $C$ at points $A$ and $B$. If $\angle AMB = 90 ^ { \circ }$, then $k = $ $\_\_\_\_$.

The line $x - 2y + 1 = 0$ intersects the parabola $y^{2} = 2px \ (p > 0)$ at points $A , B$ with $AB = 4\sqrt{15}$ .
(1) Find the value of $p$ ;
(2) Let $F$ be the focus of $y^{2} = 2px$ . Let $M , N$ be two points on the parabola such that $\overrightarrow{MF} \perp \overrightarrow{NF}$ . Find the minimum area of $\triangle MNF$ .

Let $O$ be the origin of coordinates. The line $y=-\sqrt{3}(x-1)$ passes through the focus of the parabola $C: y^2=2px$ $(p>0)$ and intersects $C$ at points $M$ and $N$. Let $l$ be the directrix of $C$. Then
A. $p=2$
B. $|MN|=\frac{8}{3}$
C. the circle with $MN$ as diameter is tangent to $l$
D. $\triangle OMN$ is an isosceles triangle

Let the focus of parabola $C: y^2 = 6x$ be $F$. A line through $F$ intersects $C$ at $A$ and $B$. A line through $F$ perpendicular to $AB$ intersects the directrix $l: x = -\frac{3}{2}$ at $E$. From point $A$, draw a perpendicular to the directrix $l$ with foot $D$. Then
A. $|AD| = |AF|$
B. $|AE| = |AB|$
C. $|AB| \geq 6$
D. $|AE| \cdot |BE| \geq 18$

Let the focus of parabola $C: y^2 = 6x$ be $F$. A line through $F$ intersects $C$ at $A$ and $B$. A perpendicular from $A$ to the line $l: x = -\frac{3}{2}$ meets it at $D$. A line through $F$ perpendicular to $AB$ meets $l$ at $E$. Then
A. $|AD| = |AF|$
B. $|AE| = |AB|$
C. $|AB| \geq 6$
D. $|AE| \cdot |BE| \geq 18$

Let $P Q$ be a focal chord of the parabola $y ^ { 2 } = 4 a x$. The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y = 2 x + a , a > 0$.
Length of chord $P Q$ is
(A) $7 a$
(B) $5 a$
(C) $2 a$
(D) $3 a$

Let $P Q$ be a focal chord of the parabola $y ^ { 2 } = 4 a x$. The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y = 2 x + a , a > 0$.
If chord $P Q$ subtends an angle $\theta$ at the vertex of $y ^ { 2 } = 4 a x$, then $\tan \theta =$
(A) $\frac { 2 } { 3 } \sqrt { 7 }$
(B) $\frac { - 2 } { 3 } \sqrt { 7 }$
(C) $\frac { 2 } { 3 } \sqrt { 5 }$
(D) $\frac { - 2 } { 3 } \sqrt { 5 }$

A chord is drawn through the focus of the parabola $y ^ { 2 } = 6 x$ such that its distance from the vertex of this parabola is $\frac { \sqrt { 5 } } { 2 }$, then its slope can be
(1) $\frac { \sqrt { 5 } } { 2 }$
(2) $\frac { 2 } { \sqrt { 3 } }$
(3) $\frac { \sqrt { 3 } } { 2 }$
(4) $\frac { 2 } { \sqrt { 5 } }$

If one end of a focal chord of the parabola, $y ^ { 2 } = 16 x$ is at $( 1,4 )$, then the length of this focal chord is
(1) 24
(2) 25
(3) 22
(4) 20

If one end of a focal chord $AB$ of the parabola $y ^ { 2 } = 8 x$ is at $A \left( \frac { 1 } { 2 } , - 2 \right)$, then the equation of the tangent to it at $B$ is:
(1) $2 x + y - 24 = 0$
(2) $x - 2 y + 8 = 0$
(3) $x + 2 y + 8 = 0$
(4) $2 x - y - 24 = 0$

Let $P Q$ be a focal chord of the parabola $y ^ { 2 } = 4 x$ such that it subtends an angle of $\frac { \pi } { 2 }$ at the point $( 3,0 )$. Let the line segment $P Q$ be also a focal chord of the ellipse $E : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 , a ^ { 2 } > b ^ { 2 }$. If $e$ is the eccentricity of the ellipse $E$, then the value of $\frac { 1 } { e ^ { 2 } }$ is equal to
(1) $1 + \sqrt { 2 }$
(2) $3 + 2 \sqrt { 2 }$
(3) $1 + 2 \sqrt { 3 }$
(4) $4 + 5 \sqrt { 3 }$

Let $R$ be the focus of the parabola $y ^ { 2 } = 20 x$ and the line $y = m x + c$ intersect the parabola at two points $P$ and $Q$. Let the point $G ( 10 , 10 )$ be the centroid of the triangle $P Q R$. If $c - m = 6$, then $P Q ^ { 2 }$ is
(1) 296
(2) 325
(3) 317
(4) 346

Q82. Let the length of the focal chord PQ of the parabola $y ^ { 2 } = 12 x$ be 15 units. If the distance of PQ from the origin is p , then $10 \mathrm { p } ^ { 2 }$ is equal to $\_\_\_\_$

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 $\angle 5 = \angle FF'B$