As shown in the figure, for a point P on the hyperbola $\frac { x ^ { 2 } } { 8 } - \frac { y ^ { 2 } } { 17 } = 1$ with foci $\mathrm { F } , \mathrm { F } ^ { \prime }$, there is a circle $C$ that is tangent to both line FP and line $\mathrm { F } ^ { \prime } \mathrm { P }$ simultaneously and has its center on the $y$-axis. For point Q, the point of tangency of line $\mathrm { F } ^ { \prime } \mathrm { P }$ with circle $C$, we have $\overline { \mathrm { F } ^ { \prime } \mathrm { Q } } = 5 \sqrt { 2 }$. Find the value of $\overline { \mathrm { FP } } ^ { 2 } + { \overline { \mathrm { F } ^ { \prime } \mathrm { P } } } ^ { 2 }$. (Here, $\overline { \mathrm { F } ^ { \prime } \mathrm { P } } < \overline { \mathrm { FP } }$) [4 points]

For an ellipse $\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 16 } = 1$ with foci $\mathrm { F } , \mathrm { F } ^ { \prime }$, there is a point A in the first quadrant on the ellipse. Among circles that are tangent to both lines $\mathrm { AF } , \mathrm { AF } ^ { \prime }$ and have their center on the y-axis, let C be the circle whose center has a negative y-coordinate. When the center of circle C is B and the area of quadrilateral $\mathrm { AFBF } ^ { \prime }$ is 72, what is the radius of circle C? [3 points]
(1) $\frac { 17 } { 2 }$
(2) 9
(3) $\frac { 19 } { 2 }$
(4) 10
(5) $\frac { 21 } { 2 }$

Let the parabola $C : y ^ { 2 } = 4 x$ have focus $F$. A line $l$ through $F$ with slope $k ( k > 0 )$ intersects $C$ at points $A , B$, with $| A B | = 8$.
(1) Find the equation of $l$;
(2) Find the equation of the circle passing through points $A , B$ and tangent to the directrix of $C$.

(12 points)
Let the focus of parabola $C : y ^ { 2 } = 4 x$ be $F$. A line $l$ passing through $F$ with slope $k ( k > 0 )$ intersects $C$ at points $A$ and $B$. $| A B | = 8$.
(1) Find the equation of line $l$;
(2) Find the equation of the circle passing through points $A$ and $B$ and tangent to the directrix of $C$.

The directrix of parabola $C : y ^ { 2 } = 4 x$ is $l$. Let $P$ be a moving point on $C$. Draw a tangent line to circle $\odot A : x ^ { 2 } + ( y - 4 ) ^ { 2 } = 1$ through $P$, with $Q$ as the point of tangency. Draw a perpendicular from $P$ to line $l$, with $B$ as the foot of the perpendicular. Then
A. Line $l$ is tangent to $\odot A$
B. When $P , A , B$ are collinear, $| P Q | = \sqrt { 15 }$
C. When $| P B | = 2$, $P A \perp A B$
D. There are exactly 2 points $P$ satisfying $| P A | = | P B |$

Let A and B be two distinct points on the parabola $\mathrm { y } ^ { 2 } = 4 \mathrm { x }$. If the axis of the parabola touches a circle of radius $r$ having $A B$ as its diameter, then the slope of the line joining A and B can be
A) $- \frac { 1 } { r }$
B) $\frac { 1 } { r }$
C) $\frac { 2 } { r }$
D) $- \frac { 2 } { \mathrm { r } }$

The normal at $\left( 2 , \frac { 3 } { 2 } \right)$ to the ellipse, $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 3 } = 1$ touches a parabola, whose equation is
(1) $y ^ { 2 } = - 104 x$
(2) $y ^ { 2 } = 14 x$
(3) $y ^ { 2 } = 26 x$
(4) $y ^ { 2 } = - 14 x$

If the line $y = mx + c$ is a common tangent to the hyperbola $\frac{x^2}{100} - \frac{y^2}{64} = 1$ and the circle $x^2 + y^2 = 36$, then which one of the following is true?
(1) $c^2 = 369$
(2) $5m = 4$
(3) $4c^2 = 369$
(4) $8m + 5 = 0$

If the points of intersection of the ellipse $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ and the circle $x ^ { 2 } + y ^ { 2 } = 4b , b > 4$ lie on the curve $y ^ { 2 } = 3x ^ { 2 }$, then $b$ is equal to :
(1) 12
(2) 5
(3) 6
(4) 10

The locus of the midpoints of the chord of the circle, $x ^ { 2 } + y ^ { 2 } = 25$ which is tangent to the hyperbola, $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 16 } = 1$ is:

If a tangent to the ellipse $x ^ { 2 } + 4 y ^ { 2 } = 4$ meets the tangents at the extremities of its major axis at $B$ and $C$, then the circle with $B C$ as diameter passes through the point.
(1) $( \sqrt { 3 } , 0 )$
(2) $( \sqrt { 2 } , 0 )$
(3) $( 1,1 )$
(4) $( - 1,1 )$

Let a circle of radius 4 be concentric to the ellipse $15 x ^ { 2 } + 19 y ^ { 2 } = 285$. Then the common tangents are inclined to the minor axis of the ellipse at the angle
(1) $\frac { \pi } { 3 }$
(2) $\frac { \pi } { 4 }$
(3) $\frac { \pi } { 6 }$
(4) $\frac { \pi } { 12 }$

Consider a hyperbola H having centre at the origin and foci on the x-axis. Let $\mathrm { C } _ { 1 }$ be the circle touching the hyperbola H and having the centre at the origin. Let $\mathrm { C } _ { 2 }$ be the circle touching the hyperbola H at its vertex and having the centre at one of its foci. If areas (in sq units) of $C _ { 1 }$ and $C _ { 2 }$ are $36 \pi$ and $4 \pi$, respectively, then the length (in units) of latus rectum of H is
(1) $\frac { 14 } { 3 }$
(2) $\frac { 28 } { 3 }$
(3) $\frac { 11 } { 3 }$
(4) $\frac { 10 } { 3 }$

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) $9 x - 9 y = 32$

Q67. Consider a hyperbola H having centre at the origin and foci on the x -axis. Let $\mathrm { C } _ { 1 }$ be the circle touching the hyperbola H and having the centre at the origin. Let $\mathrm { C } _ { 2 }$ be the circle touching the hyperbola H at its vertex and having the centre at one of its foci. If areas (in sq units) of $C _ { 1 }$ and $C _ { 2 }$ are $36 \pi$ and $4 \pi$, respectively, then the length (in units) of latus rectum of H is
(1) $\frac { 14 } { 3 }$
(2) $\frac { 28 } { 3 }$
(3) $\frac { 11 } { 3 }$
(4) $\frac { 10 } { 3 }$