The complex plane is equipped with a direct orthonormal coordinate system ($\mathrm{O}; \vec{u}, \vec{v}$). We denote by $\mathscr{C}$ the set of points $M$ in the plane with affix $z$ such that $|z - 2| = 1$.

1. Justify that $\mathscr{C}$ is a circle, and specify its center and radius.
2. Let $a$ be a real number. We call $\mathscr{D}$ the line with equation $y = ax$. Determine the number of intersection points between $\mathscr{C}$ and $\mathscr{D}$ as a function of the values of the real number $a$.

For two points $\mathrm { A } ( 1 , \sqrt { 3 } ) , \mathrm { B } ( 1 , - \sqrt { 3 } )$ on the coordinate plane, what is the total length of the figure represented by point $\mathrm { P } ( x , y )$ satisfying the following two conditions? [4 points] (가) $x ^ { 2 } + y ^ { 2 } = 4$ (나) For any point $( 1 , a )$ on segment AB, the matrix $\left( \begin{array} { c c } x & y \\ 1 & a \end{array} \right)$ has an inverse matrix.
(1) $\frac { 1 } { 3 } \pi$
(2) $\frac { 1 } { 2 } \pi$
(3) $\pi$
(4) $\frac { 4 } { 3 } \pi$
(5) $\frac { 3 } { 2 } \pi$

On the coordinate plane, for two points $\mathrm { A } ( 1 , \sqrt { 3 } ) , \mathrm { B } ( 1 , - \sqrt { 3 } )$, what is the total length of the figure represented by points $\mathrm { P } ( x , y )$ satisfying the following two conditions? [4 points]
(가) $x ^ { 2 } + y ^ { 2 } = 4$
(나) For any point $( 1 , a )$ on the line segment AB, the matrix $\left( \begin{array} { l l } x & y \\ 1 & a \end{array} \right)$ has an inverse matrix.
(1) $\frac { 1 } { 3 } \pi$
(2) $\frac { 1 } { 2 } \pi$
(3) $\pi$
(4) $\frac { 4 } { 3 } \pi$
(5) $\frac { 3 } { 2 } \pi$

20. (12 points) The ellipse $\mathrm { E } : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ has semi-focal distance $c$. The distance from the origin O to the line passing through the points $( c , 0 )$ and $( 0 , b )$ is $\frac { 1 } { 2 } c$. (I) Find the eccentricity of ellipse E; (II) As shown in the figure, AB is a diameter of circle $\mathrm { M } : ( x + 2 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = \frac { 5 } { 2 }$. If the ellipse E passes through points A and B, find the equation of ellipse E. [Figure]

10. Let line $l$ intersect the parabola $y ^ { 2 } = 4 x$ at points $A , B$, and be tangent to the circle $( x - 5 ) ^ { 2 } + y ^ { 2 } = r ^ { 2 } ( r > 0 )$ at point $M$. If $M$ is the midpoint of segment $A B$, and there are exactly 4 such lines $l$, then the range of $r$ is
(A) $( 1,3 )$
(B) $( 1,4 )$
(C) $( 2,3 )$
(D) $( 2,4 )$
II. Fill in the Blanks:

10. Let line $l$ intersect the parabola $y ^ { 2 } = 4 x$ at points $\mathrm { A }$ and $\mathrm { B }$, and be tangent to the circle $( x - 5 ) ^ { 2 } + y ^ { 2 } = r ^ { 2 }$ $(r > 0)$ at point $M$, where $M$ is the midpoint of segment $A B$. If there are exactly $4$ such lines $l$, then the range of $r$ is
(A) $( 1, 3 )$
(B) $( 1, 4 )$
(C) $( 2, 3 )$
(D) $( 2, 4 )$
II. Fill in the Blanks

If the circle $x^2 + (y+2)^2 = r^2$ $(r > 0)$ has exactly $2$ points at distance $1$ from the line $y = \sqrt{3}x + 2$, then the range of $r$ is
A. $(0,1)$
B. $(1,3)$
C. $(3, +\infty)$
D. $(0, +\infty)$

If the circle $x^2 + (y+2)^2 = r^2$ $(r > 0)$ has exactly 2 points at distance 1 from the line $y = \sqrt{3}x + 2$, then the range of $r$ is
A. $(0,1)$
B. $(1,3)$
C. $(3, +\infty)$
D. $(0, +\infty)$

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$, with $\varphi_A(x,y)$ the determinant of $A(x,y) = \left(\begin{array}{cc} a-x & b-y \\ c+y & d-x \end{array}\right)$ and $\mathcal{CP}_A$ the eigenvalue circle of $A$, specify, as a function of $A$, the cardinality of the intersection of $\mathcal{CP}_A$ with the $x$-axis $\mathbb{R} \times \{0\}$.

Consider points of the form $\left(n, n^k\right)$, where $n$ and $k$ are integers with $n \geq 0$, $k \geq 1$. How many such points are strictly inside the circle of radius 10 with centre at the origin?
(A) 11
(B) 12
(C) 15
(D) 17

For how many values of $p$, the circle $x^2 + y^2 + 2x + 4y - p = 0$ and the coordinate axes have exactly three common points?

If the point $( 1,4 )$ lies inside the circle $x ^ { 2 } + y ^ { 2 } - 6 x + 10 y + p = 0$ and the circle does not touch or intersect the coordinate axes, then the set of all possible values of $p$ is the interval
(1) $( 25,39 )$
(2) $( 25,29 )$
(3) $( 0,25 )$
(4) $( 9,25 )$

Let the circle $S : 36 x ^ { 2 } + 36 y ^ { 2 } - 108 x + 120 y + C = 0$ be such that it neither intersects nor touches the coordinate axes. If the point of intersection of the lines, $x - 2 y = 4$ and $2 x - y = 5$ lies inside the circle $S$, then:
(1) $\frac { 25 } { 9 } < C < \frac { 13 } { 3 }$
(2) $100 < C < 165$
(3) $81 < C < 156$
(4) $100 < C < 156$

If the variable line $3 x + 4 y = \alpha$ lies between the two circles $( x - 1 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$ and $( x - 9 ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 4$, without intercepting a chord on either circle, then the sum of all the integral values of $\alpha$ is

The number of integral values of $k$ for which the line $3x + 4y = k$ intersects the circle $x^2 + y^2 - 2x - 4y + 4 = 0$ at two distinct points is $\_\_\_\_$.

In the rectangular coordinate plane, the line $y = m x$
$$x ^ { 2 } - 26 x + y ^ { 2 } + 144 = 0$$
intersects the circle at two different points.
Accordingly, which of the following is the interval showing all possible values of $m$?
A) $\left( - \frac { 3 } { 4 } , \frac { 3 } { 4 } \right)$
B) $\left( - \frac { 3 } { 8 } , \frac { 3 } { 8 } \right)$
C) $\left( - \frac { 4 } { 9 } , \frac { 4 } { 9 } \right)$
D) $\left( - \frac { 5 } { 12 } , \frac { 5 } { 12 } \right)$
E) $\left( - \frac { 7 } { 24 } , \frac { 7 } { 24 } \right)$