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 $( x - 3 ) ^ { 2 } + ( y - 4 ) ^ { 2 } = 16$: $\_\_\_\_$ .

130- Two circles with radii 4 and 5 are externally tangent. From the center of the smaller circle, a common external tangent to the larger circle is drawn. What is the length of this tangent segment?
(1) $8$ (2) $4\sqrt{5}$ (3) $4\sqrt{6}$ (4) $15$

135. For which value of $a$, the angle between the tangent line to the circle $x^2 + y^2 - 2x + y = 1$ and the line $3x + 2y = a$ at their intersection point is $90°$?
(1) $2$ (2) $3$ (3) $4$ (4) $5$

129. In the figure below, $AD$ is tangent to the circle with center $O$, and $OH$ is perpendicular to $AC$. If $\widehat{DBC} = 2\widehat{DAC}$, how many times is angle $\widehat{COH}$ equal to angle $\widehat{DAC}$?
\begin{minipage}{0.45\textwidth} [Figure: Circle with center $O$, tangent line $AD$, points $B$, $C$, $H$ marked] \end{minipage} \begin{minipage}{0.45\textwidth} \begin{flushright} (1) $2.5$
(2) $3$
(3) $3.5$
(4) $4$ \end{flushright} \end{minipage}

152. In the figure below, line segment $AC$ is tangent to the circle. If $\dfrac{AC}{BC} = \sqrt{3}$, then what is $\dfrac{DB}{BC}$?
[Figure: Circle with points A, B, C, D, where AC is tangent to the circle at C]

• [(1)] $\sqrt{2}$
• [(2)] $\sqrt{3}$
• [(3)] $2$
• [(4)] $3$

30. In the figure below, two tangent lines are drawn from point $A$. What is the radius of the circle?
[Figure: Two tangent lines drawn from external point $A$ to a circle, with segments labeled 9, 8, 7 and point $B$, $C$ marked]

• [(1)] $7/2\sqrt{2}$
• [(2)] $4/8\sqrt{5}$
• [(3)] $3/6\sqrt{2}$
• [(4)] $2/4\sqrt{5}$

5. The number of common tangents to the circles $x 2 + y 2 = 4$ and $x 2 + y 2 - 6 x - y = 24$ is :
(A) 0
(B) 1
(C) 3
(D) 4

30. On the ellipse $4 x 2 + 9 y 2 = 1$, the points at which the tangents are parallel to the line $8 \mathrm { x } = 9 \mathrm { y }$ are :
(A) $( 2 / 5,1 / 5 )$
B) $( - 2 / 5,1 / 5 )$
(C) $( - 2 / 5 , - 1 / 5 )$
(D) $( 2 / 5 , - 1 / 5 )$

5. Let $\mathrm { T } 1 , \mathrm {~T} 2$ be two tangents drawn from $( - 2,0 )$ onto the circle $\mathrm { C } : \mathrm { x } 2 + \mathrm { y } 2 = 1$. Determine the circles touching C and having T1, T2 as their pair of tangents. Further, find the equations of all possible common tangents to these circles, when taken two at a time.

4. Let $2 \times 2 + \underset { 2 } { 2 } - 3 \times y = 0$ be the equation of a pair of tangents drawn from the origin O to a circle of radius 3 with centre in the first quadrant. If A is one of the points of contact, find the length of OA .

10. The equation of the common tangent touching the circle $( x - 3 ) 2 + y 2 = 9$ and the parabola $y 2 = 4 x$ above the $x$-axis is :
(A) $\sqrt { } 3 y = 3 x + 1$
(B) $\sqrt { } 3 y = - ( x + 3 )$
(C) $\sqrt { } 3 y = x + 3$
(D) $\sqrt { } 3 y = - ( 3 x + 1 )$

31. Let $P Q$ and $R S$ be tangents at the extremities of the diameter $P R$ of a circle of radius $r$. If PS and RQ intersect at a point X on the circumference of the circle, then 2 r equals :
(A) $\sqrt { } ( \mathrm { PQ } . \mathrm { RS } )$
(B) $( P Q + R S ) / 2$
(C) $( 2 \mathrm { PQ } \cdot \mathrm { RS } ) / ( \mathrm { PQ } + \mathrm { RS } )$
(D) $\sqrt { } ( ( P Q 2 + R S 2 ) / 2 )$

6. Prove that, in an ellipse, the perpendicular from a focus upon any tangent and the line joininɛ centre of the ellipse of the point of contact meet on the corresponding directrix.

15. If the tangent at the point $P$ on the circle $x ^ { 2 } + y ^ { 2 } + 6 x + 6 y = 2$ meets the straight line $5 x + 2 y = 6$ at a point $Q$ on the $y$-axis, then the length of $P Q$ is
(A) 4
(B) $2 \sqrt { } 5$
(C) 5
(D) $3 \sqrt { 5 }$

16. If $a > 2 b > 0$ then the positive value of $m$ for which $y = m x - b \sqrt { } \left( 1 + m ^ { 2 } \right)$ is $a$ common tangent to $x ^ { 2 } + y ^ { 2 } = b ^ { 2 }$ and $( x - a ) ^ { 2 } + y ^ { 2 } = b ^ { 2 }$ is
(A) $2 b / \sqrt { } \left( a ^ { 2 } - 4 b ^ { 2 } \right)$
(B) $\quad \sqrt { } \left( a ^ { 2 } - 4 b ^ { 2 } \right) / 2 b$

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(C) $2 \mathrm {~b} / ( \mathrm { a } - 2 \mathrm {~b} )$
(D) $\mathrm { b } / ( \mathrm { a } - 2 \mathrm {~b} )$