Fill in the blanks. Let $C_1$ be the circle with center $(-8, 0)$ and radius 6. Let $C_2$ be the circle with center $(8, 0)$ and radius 2. Given a point $P$ outside both circles, let $\ell_i(P)$ be the length of a tangent segment from $P$ to circle $C_i$. The locus of all points $P$ such that $\ell_1(P) = 3\ell_2(P)$ is a circle with radius \_\_\_ and center at (\_\_\_, \_\_\_).

In the coordinate plane, two lines $l _ { 1 } , l _ { 2 }$ tangent to the parabola $y ^ { 2 } = 8 x$ have slopes $m _ { 1 } , m _ { 2 }$ respectively. When $m _ { 1 } , m _ { 2 }$ are the two distinct roots of the equation $2 x ^ { 2 } - 3 x + 1 = 0$, what is the $x$-coordinate of the intersection point of $l _ { 1 }$ and $l _ { 2 }$? [3 points]
(1) 1
(2) 2
(3) 3
(4) 4
(5) 5

For the ellipse $\frac{x^2}{a^2} + \frac{y^2}{6} = 1$, what is the slope of the tangent line at the point $(\sqrt{3}, -2)$ on the ellipse? (where $a$ is a positive number) [3 points]
(1) $\sqrt{3}$
(2) $\frac{\sqrt{3}}{2}$
(3) $\frac{\sqrt{3}}{3}$
(4) $\frac{\sqrt{3}}{4}$
(5) $\frac{\sqrt{3}}{5}$

8. The line $3 \mathrm { x } + 4 \mathrm { y } = \mathrm { b }$ is tangent to the circle $x ^ { 2 } + y ^ { 2 } - 2 x - 2 y + 1 = 0$. Then $\mathrm { b } =$
(A) $-2$ or $12$
(B) $2$ or $-12$
(C) $-2$ or $-12$
(D) $2$ or $12$

If point $\mathrm { P } ( 1,2 )$ lies on a circle centered at the origin, then the equation of the tangent line to the circle at point $P$ is $\_\_\_\_$ .

8. Given that the line $l$: $x + a y - 1 = 0 ( a \in R )$ is an axis of symmetry of the circle $C$: $x ^ { 2 } + y ^ { 2 } - 4 x - 2 y + 1 = 0$. A tangent line to circle $C$ is drawn from point $\mathrm { A } ( - 4 , \mathrm { a } )$, with tangent point $B$. Then $| \mathrm { AB } | =$
A. $2$
B. $4 \sqrt { 2 }$
C. $6$
D. $2 \sqrt { 10 }$

15. (Elective 4-1: Geometric Proof) As shown in the figure, $PA$ is tangent to the circle at point $A$, and $PBC$ is a secant line with $BC = 3PB$. Then $\frac{AB}{AC} = $ $\_\_\_\_$ .

19. (15 points) As shown in the figure, given the parabola $\mathrm { C } _ { 1 } : \mathrm { y } = \frac { 1 } { 4 } x ^ { 2 }$ , the circle $\mathrm { C } _ { 2 } : x ^ { 2 } + ( \mathrm { y } - 1 ) ^ { 2 } = 1$ , through point $\mathrm { P } ( \mathrm { t } , 0 ) ( \mathrm { t } > 0 )$ , draw lines $\mathrm { PA } , \mathrm { PB}$ not passing through the origin O that are tangent to the parabola $C _ { 1 }$ and circle $\mathrm { C } _ { 2 }$ respectively, with $\mathrm { A } , \mathrm { B}$ as the points of tangency.
(1) Find the coordinates of points $\mathrm { A } , \mathrm { B}$ ;
(2) Find the area of $\triangle \mathrm { PAB}$ . Note: If a line has exactly one common point with a parabola and is not parallel to the axis of symmetry of the parabola, then the line is tangent to the parabola, and the common point is called the point of tangency. [Figure]

21. The parabola $C$ has its vertex at the origin $O$ and its focus on the $x$-axis. The line $l : x = 1$ intersects $C$ at points $P , Q$, and $O P \perp O Q$. Given the point $M ( 2,0 )$, and circle $\odot M$ is tangent to $l$.
(1) Find the equations of $C$ and $\odot M$;
(2) Let $A _ { 1 } , A _ { 2 } , A _ { 3 }$ be three points on $C$. Lines $A _ { 1 } A _ { 2 }$ and $A _ { 1 } A _ { 3 }$ are both tangent to $\odot M$. Determine the positional relationship between line $A _ {

20. (12 points) The parabola $C$ has its vertex at the origin $O$ and focus on the $x$-axis. The line $l: x =

14. Write the equation of a line that is tangent to both the circle $x ^ { 2 } + y ^ { 2 } = 1$ and the circle $( x - 3 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 16$: $\_\_\_\_$ .

The circle $x ^ { 2 } + y ^ { 2 } - 8 x = 0$ and hyperbola $\frac { x ^ { 2 } } { 9 } - \frac { y ^ { 2 } } { 4 } = 1$ intersect at the points $A$ and $B$.
Equation of a common tangent with positive slope to the circle as well as to the hyperbola is
A) $2 x - \sqrt { 5 } y - 20 = 0$
B) $2 x - \sqrt { 5 } y + 4 = 0$
C) $3 x - 4 y + 8 = 0$
D) $4 x - 3 y + 4 = 0$

If the normals of the parabola $y ^ { 2 } = 4 x$ drawn at the end points of its latus rectum are tangents to the circle $( x - 3 ) ^ { 2 } + ( y + 2 ) ^ { 2 } = r ^ { 2 }$, then the value of $r ^ { 2 }$ is

Suppose that the foci of the ellipse $\frac { x ^ { 2 } } { 9 } + \frac { y ^ { 2 } } { 5 } = 1$ are $\left( f _ { 1 } , 0 \right)$ and $\left( f _ { 2 } , 0 \right)$ where $f _ { 1 } > 0$ and $f _ { 2 } < 0$. Let $P _ { 1 }$ and $P _ { 2 }$ be two parabolas with a common vertex at ( 0,0 ) and with foci at ( $f _ { 1 } , 0$ ) and ( $2 f _ { 2 } , 0$ ), respectively. Let $T _ { 1 }$ be a tangent to $P _ { 1 }$ which passes through ( $2 f _ { 2 } , 0$ ) and $T _ { 2 }$ be a tangent to $P _ { 2 }$ which passes through $\left( f _ { 1 } , 0 \right)$. If $m _ { 1 }$ is the slope of $T _ { 1 }$ and $m _ { 2 }$ is the slope of $T _ { 2 }$, then the value of $\left( \frac { 1 } { m _ { 1 } ^ { 2 } } + m _ { 2 } ^ { 2 } \right)$ is

Let $E _ { 1 }$ and $E _ { 2 }$ be two ellipses whose centers are at the origin. The major axes of $E _ { 1 }$ and $E _ { 2 }$ lie along the $x$-axis and the $y$-axis, respectively. Let $S$ be the circle $x ^ { 2 } + ( y - 1 ) ^ { 2 } = 2$. The straight line $x + y = 3$ touches the curves $S , E _ { 1 }$ and $E _ { 2 }$ at $P , Q$ and $R$, respectively. Suppose that $P Q = P R = \frac { 2 \sqrt { 2 } } { 3 }$. If $e _ { 1 }$ and $e _ { 2 }$ are the eccentricities of $E _ { 1 }$ and $E _ { 2 }$, respectively, then the correct expression(s) is(are)
(A) $e _ { 1 } ^ { 2 } + e _ { 2 } ^ { 2 } = \frac { 43 } { 40 }$
(B) $\quad e _ { 1 } e _ { 2 } = \frac { \sqrt { 7 } } { 2 \sqrt { 10 } }$
(C) $\left| e _ { 1 } ^ { 2 } - e _ { 2 } ^ { 2 } \right| = \frac { 5 } { 8 }$
(D) $e _ { 1 } e _ { 2 } = \frac { \sqrt { 3 } } { 4 }$

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

	Column 1	Column 2	Column 3
	(I) $x^2 + y^2 = a^2$	(i) $my = m^2x + a$	(P) $\left(\frac{a}{m^2}, \frac{2a}{m}\right)$
(II)	$x^2 + a^2y^2 = a^2$	(ii) $y = mx + a\sqrt{m^2+1}$	(Q) $\left(\frac{-ma}{\sqrt{m^2+1}}, \frac{a}{\sqrt{m^2+1}}\right)$
(III)	$y^2 = 4ax$	(iii) $y = mx + \sqrt{a^2m^2 - 1}$	(R) $\left(\frac{-a^2m}{\sqrt{a^2m^2+1}}, \frac{1}{\sqrt{a^2m^2+1}}\right)$
(IV)	$x^2 - a^2y^2 = a^2$	(iv) $y = mx + \sqrt{a^2m^2+1}$	(S) $\left(\frac{-a^2m}{\sqrt{a^2m^2-1}}, \frac{-1}{\sqrt{a^2m^2-1}}\right)$

For $a = \sqrt{2}$, if a tangent is drawn to a suitable conic (Column 1) at the point of contact $(-1, 1)$, then which of the following options is the only CORRECT combination for obtaining its equation?
[A] (I) (i) (P)
[B] (I) (ii) (Q)
[C] (II) (ii) (Q)
[D] (III) (i) (P)

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

	Column 1	Column 2	Column 3
	(I) $x^2 + y^2 = a^2$	(i) $my = m^2x + a$	(P) $\left(\frac{a}{m^2}, \frac{2a}{m}\right)$
(II)	$x^2 + a^2y^2 = a^2$	(ii) $y = mx + a\sqrt{m^2+1}$	(Q) $\left(\frac{-ma}{\sqrt{m^2+1}}, \frac{a}{\sqrt{m^2+1}}\right)$
(III)	$y^2 = 4ax$	(iii) $y = mx + \sqrt{a^2m^2 - 1}$	(R) $\left(\frac{-a^2m}{\sqrt{a^2m^2+1}}, \frac{1}{\sqrt{a^2m^2+1}}\right)$
(IV)	$x^2 - a^2y^2 = a^2$	(iv) $y = mx + \sqrt{a^2m^2+1}$	(S) $\left(\frac{-a^2m}{\sqrt{a^2m^2-1}}, \frac{-1}{\sqrt{a^2m^2-1}}\right)$

If a tangent to a suitable conic (Column 1) is found to be $y = x + 8$ and its point of contact is $(8, 16)$, then which of the following options is the only CORRECT combination?
[A] (I) (ii) (Q)
[B] (II) (iv) (R)
[C] (III) (i) (P)
[D] (III) (ii) (Q)

Consider two straight lines, each of which is tangent to both the circle $x ^ { 2 } + y ^ { 2 } = \frac { 1 } { 2 }$ and the parabola $y ^ { 2 } = 4 x$. Let these lines intersect at the point $Q$. Consider the ellipse whose center is at the origin $O ( 0,0 )$ and whose semi-major axis is $O Q$. If the length of the minor axis of this ellipse is $\sqrt { 2 }$, then which of the following statement(s) is (are) TRUE?
(A) For the ellipse, the eccentricity is $\frac { 1 } { \sqrt { 2 } }$ and the length of the latus rectum is 1
(B) For the ellipse, the eccentricity is $\frac { 1 } { 2 }$ and the length of the latus rectum is $\frac { 1 } { 2 }$
(C) The area of the region bounded by the ellipse between the lines $x = \frac { 1 } { \sqrt { 2 } }$ and $x = 1$ is $\frac { 1 } { 4 \sqrt { 2 } } ( \pi - 2 )$
(D) The area of the region bounded by the ellipse between the lines $x = \frac { 1 } { \sqrt { 2 } }$ and $x = 1$ is
$$\frac { 1 } { 16 } ( \pi - 2 )$$

Let the point $B$ be the reflection of the point $A ( 2,3 )$ with respect to the line $8 x - 6 y - 23 = 0$. Let $\Gamma _ { A }$ and $\Gamma _ { B }$ be circles of radii 2 and 1 with centres $A$ and $B$ respectively. Let $T$ be a common tangent to the circles $\Gamma _ { A }$ and $\Gamma _ { B }$ such that both the circles are on the same side of $T$. If $C$ is the point of intersection of $T$ and the line passing through $A$ and $B$, then the length of the line segment $A C$ is $\_\_\_\_$

Let $E$ denote the parabola $y ^ { 2 } = 8 x$. Let $P = ( - 2,4 )$, and let $Q$ and $Q ^ { \prime }$ be two distinct points on $E$ such that the lines $P Q$ and $P Q ^ { \prime }$ are tangents to $E$. Let $F$ be the focus of $E$. Then which of the following statements is (are) TRUE ?
(A) The triangle $P F Q$ is a right-angled triangle
(B) The triangle $Q P Q ^ { \prime }$ is a right-angled triangle
(C) The distance between $P$ and $F$ is $5 \sqrt { 2 }$
(D) $F$ lies on the line joining $Q$ and $Q ^ { \prime }$

Consider the parabola $y ^ { 2 } = 4 x$. Let $S$ be the focus of the parabola. A pair of tangents drawn to the parabola from the point $P = ( - 2,1 )$ meet the parabola at $P _ { 1 }$ and $P _ { 2 }$. Let $Q _ { 1 }$ and $Q _ { 2 }$ be points on the lines $S P _ { 1 }$ and $S P _ { 2 }$ respectively such that $P Q _ { 1 }$ is perpendicular to $S P _ { 1 }$ and $P Q _ { 2 }$ is perpendicular to $S P _ { 2 }$. Then, which of the following is/are TRUE?
(A) $\quad S Q _ { 1 } = 2$
(B) $\quad Q _ { 1 } Q _ { 2 } = \frac { 3 \sqrt { 10 } } { 5 }$
(C) $\quad P Q _ { 1 } = 3$
(D) $\quad S Q _ { 2 } = 1$

Let $T _ { 1 }$ and $T _ { 2 }$ be two distinct common tangents to the ellipse $E : \frac { x ^ { 2 } } { 6 } + \frac { y ^ { 2 } } { 3 } = 1$ and the parabola $P : y ^ { 2 } = 12 x$. Suppose that the tangent $T _ { 1 }$ touches $P$ and $E$ at the points $A _ { 1 }$ and $A _ { 2 }$, respectively and the tangent $T _ { 2 }$ touches $P$ and $E$ at the points $A _ { 4 }$ and $A _ { 3 }$, respectively. Then which of the following statements is(are) true?
(A) The area of the quadrilateral $A _ { 1 } A _ { 2 } A _ { 3 } A _ { 4 }$ is 35 square units
(B) The area of the quadrilateral $A _ { 1 } A _ { 2 } A _ { 3 } A _ { 4 }$ is 36 square units
(C) The tangents $T _ { 1 }$ and $T _ { 2 }$ meet the $x$-axis at the point $( - 3,0 )$
(D) The tangents $T _ { 1 }$ and $T _ { 2 }$ meet the $x$-axis at the point $( - 6,0 )$

Let $C _ { 1 }$ be the circle of radius 1 with center at the origin. Let $C _ { 2 }$ be the circle of radius $r$ with center at the point $A = ( 4,1 )$, where $1 < r < 3$. Two distinct common tangents $P Q$ and $S T$ of $C _ { 1 }$ and $C _ { 2 }$ are drawn. The tangent $P Q$ touches $C _ { 1 }$ at $P$ and $C _ { 2 }$ at $Q$. The tangent $S T$ touches $C _ { 1 }$ at $S$ and $C _ { 2 }$ at $T$. Mid points of the line segments $P Q$ and $S T$ are joined to form a line which meets the $x$-axis at a point $B$. If $A B = \sqrt { 5 }$, then the value of $r ^ { 2 }$ is

Let $A _ { 1 } , B _ { 1 } , C _ { 1 }$ be three points in the $xy$-plane. Suppose that the lines $A _ { 1 } C _ { 1 }$ and $B _ { 1 } C _ { 1 }$ are tangents to the curve $y ^ { 2 } = 8 x$ at $A _ { 1 }$ and $B _ { 1 }$, respectively. If $O = ( 0,0 )$ and $C _ { 1 } = ( - 4,0 )$, then which of the following statements is (are) TRUE?
(A) The length of the line segment $OA _ { 1 }$ is $4 \sqrt { 3 }$
(B) The length of the line segment $A _ { 1 } B _ { 1 }$ is 16
(C) The orthocenter of the triangle $A _ { 1 } B _ { 1 } C _ { 1 }$ is $( 0,0 )$
(D) The orthocenter of the triangle $A _ { 1 } B _ { 1 } C _ { 1 }$ is $( 1,0 )$

Let $S$ denote the locus of the point of intersection of the pair of lines
$$\begin{gathered} 4 x - 3 y = 12 \alpha \\ 4 \alpha x + 3 \alpha y = 12 \end{gathered}$$
where $\alpha$ varies over the set of non-zero real numbers. Let $T$ be the tangent to $S$ passing through the points $( p , 0 )$ and $( 0 , q ) , q > 0$, and parallel to the line $4 x - \frac { 3 } { \sqrt { 2 } } y = 0$.
Then the value of $p q$ is

(A)

$- 6 \sqrt { 2 }$

(B)

$- 3 \sqrt { 2 }$

(C)

$- 9 \sqrt { 2 }$

(D)

$- 12 \sqrt { 2 }$