For the ellipse $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 1$, a line $l$ passes through the right focus $F$ and intersects the ellipse at points $P$ and $Q$, with $P$ in the second quadrant. Given $Q \left( x _ { Q } , y _ { Q } \right)$ and $Q ^ { \prime } \left( x _ { Q } ^ { \prime } , y _ { Q } ^ { \prime } \right)$ both on the ellipse, with $y _ { Q } + y _ { Q } ^ { \prime } = 0$ and $F Q ^ { \prime } \perp P Q$, find the equation of line $l$ as $\_\_\_\_$

(1) Find the equation of $C$;
(2) Let the lines $MD$ and $ND$ intersect $C$ at another point $A$ and $B$ respectively. Denote the inclination angles of lines $MN$ and $AB$ as $\alpha$ and $\beta$ respectively. When $\alpha - \beta$ attains its maximum value, find the equation of line $AB$.

Let $A$ and $B$ be two points on the hyperbola $x ^ { 2 } - \frac { y ^ { 2 } } { 9 } = 1$. Which of the following four points could be the midpoint of segment $AB$?
A. $(1,1)$
B. $( - 1,2 )$
C. $( 1,3 )$
D. $( - 1 , - 4 )$

Given the ellipse $\frac{x^2}{3}+y^2=1$ with left and right foci $F_1, F_2$ respectively, the line $y=x+m$ intersects $C$ at points $A$ and $B$. If the area of $\triangle F_1AB$ is 2 times the area of $\triangle F_2AB$, then $m=$
A. $\frac{2}{3}$
B. $\frac{\sqrt{2}}{3}$
C. $-\frac{\sqrt{2}}{3}$
D. $-\frac{2}{3}$

Given the ellipse equation $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$. The foci and endpoints of the minor axis form a square with side length 2. A line $l$ passing through $( 0 , t ) ( t > \sqrt { 2 })$ intersects the ellipse at points $A, B$, and $C ( 0,1 )$. Connect $AC$ and it intersects the ellipse at $D$.
(1) Find the equation of the ellipse and its eccentricity;
(2) If the slope of line $BD$ is 0, find $t$.

Let a line with slope of $60 ^ { \circ }$ be drawn through the focus $F$ of the parabola $y ^ { 2 } = 8 ( x + 2 )$. If the two points of intersection of the line with the parabola are $A$ and $B$ and the perpendicular bisector of the chord $A B$ intersects the $x$-axis at the point $P$, then the length of the segment PF is
(a) $16 / 3$
(b) $8 / 3$
(c) $16 \sqrt{3} / 3$
(d) $8 \sqrt{3}$

The chord $PQ$ of the parabola $y ^ { 2 } = x$, where one end $P$ of the chord is at point $( 4 , - 2 )$, is perpendicular to the axis of the parabola. Then the slope of the normal at $Q$ is
(1) $-4$
(2) $- \frac { 1 } { 4 }$
(3) $4$
(4) $\frac { 1 } { 4 }$

If $P_{1}$ and $P_{2}$ are two points on the ellipse $\frac{x^{2}}{4} + y^{2} = 1$ at which the tangents are parallel to the chord joining the points $(0, 1)$ and $(2, 0)$, then the distance between $P_{1}$ and $P_{2}$ is
(1) $2\sqrt{2}$
(2) $\sqrt{5}$
(3) $2\sqrt{3}$
(4) $\sqrt{10}$

Let P be a point on the parabola, $y ^ { 2 } = 12 x$ and N be the foot of the perpendicular drawn from $P$, on the axis of the parabola. A line is now drawn through the mid-point $M$ of $P N$, parallel to its axis which meets the parabola at $Q$. If the $y$-intercept of the line NQ is $\frac { 4 } { 3 }$, then:
(1) $P N = 4$
(2) $M Q = \frac { 1 } { 3 }$
(3) $M Q = \frac { 1 } { 4 }$
(4) $P N = 3$

Consider the parabola with vertex $\left(\frac { 1 } { 2 } , \frac { 3 } { 4 }\right)$ and the directrix $y = \frac { 1 } { 2 }$. Let P be the point where the parabola meets the line $x = - \frac { 1 } { 2 }$. If the normal to the parabola at P intersects the parabola again at the point Q, then $(PQ) ^ { 2 }$ is equal to :
(1) $\frac { 25 } { 2 }$
(2) $\frac { 75 } { 8 }$
(3) $\frac { 125 } { 16 }$
(4) $\frac { 15 } { 2 }$

Let $P \left( \frac { 2 \sqrt { 3 } } { \sqrt { 7 } } , \frac { 6 } { \sqrt { 7 } } \right) , Q , R$ and $S$ be four points on the ellipse $9 x ^ { 2 } + 4 y ^ { 2 } = 36$. Let $P Q$ and $R S$ be mutually perpendicular and pass through the origin. If $\frac { 1 } { ( P Q ) ^ { 2 } } + \frac { 1 } { ( R S ) ^ { 2 } } = \frac { p } { q }$, where $p$ and $q$ are coprime, then $p + q$ is equal to
(1) 147
(2) 143
(3) 137
(4) 157

The length of the chord of the ellipse $\frac { x ^ { 2 } } { 25 } + \frac { y ^ { 2 } } { 16 } = 1$, whose mid point is $\left( 1 , \frac { 2 } { 5 } \right)$, is equal to:
(1) $\frac { \sqrt { 1691 } } { 5 }$
(2) $\frac { \sqrt { 2009 } } { 5 }$
(3) $\frac { \sqrt { 1741 } } { 5 }$
(4) $\frac { \sqrt { 1541 } } { 5 }$

Let $e _ { 1 }$ be the eccentricity of the hyperbola $\frac { x ^ { 2 } } { 16 } - \frac { y ^ { 2 } } { 9 } = 1$ and $e _ { 2 }$ be the eccentricity of the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 , a > b$, which passes through the foci of the hyperbola. If $e _ { 1 } e _ { 2 } = 1$, then the length of the chord of the ellipse parallel to the x -axis and passing through $( 0,2 )$ is:
(1) $4 \sqrt { 5 }$
(2) $\frac { 8 \sqrt { 5 } } { 3 }$
(3) $\frac { 10 \sqrt { 5 } } { 3 }$
(4) $3 \sqrt { 5 }$

The length of the chord of the ellipse $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 2 } = 1$, whose mid-point is $\left( 1 , \frac { 1 } { 2 } \right)$, is :
(1) $\frac { 5 } { 3 } \sqrt { 15 }$
(2) $\frac { 1 } { 3 } \sqrt { 15 }$
(3) $\frac { 2 } { 3 } \sqrt { 15 }$
(4) $\sqrt { 15 }$

If $\alpha x + \beta y = 109$ is the equation of the chord of the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 4 } = 1$, whose mid point is $\left( \frac { 5 } { 2 } , \frac { 1 } { 2 } \right)$, then $\alpha + \beta$ is equal to :
(1) 58
(2) 46
(3) 37
(4) 72

Q66. Let PQ be a chord of the parabola $y ^ { 2 } = 12 x$ and the midpoint of PQ be at $( 4,1 )$. Then, which of the following point lies on the line passing through the points P and Q ?
(1) $( 3 , - 3 )$
(2) $( 2 , - 9 )$
(3) $\left( \frac { 3 } { 2 } , - 16 \right)$
(4) $\left( \frac { 1 } { 2 } , - 20 \right)$