16. Let $F _ { 1 } , F _ { 2 }$ be the two foci of the ellipse $C : \frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1$. Let $P , Q$ be two points on $C$ that are symmetric with respect to the origin, and $| P Q | = \left| F _ { 1 } F _ { 2 } \right|$. Then the area of quadrilateral $P F _ { 1 } Q F _ { 2 }$ is $\_\_\_\_$ .

III. Solution Questions: Total 70 points. Solutions should include explanations, proofs, or calculation steps. Questions 17--21 are required questions that all students must answer. Questions 22 and 23 are optional questions; students should answer according to the requirements.

(A) Required Questions: Total 60 points.

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

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 _ {

Point $M$ lies on the line $2 x + y - 1 = 0$. Both points $( 3,0 )$ and $( 0,1 )$ lie on circle $\odot M$. Then the equation of $\odot M$ is $\_\_\_\_$ .

The equation of a circle passing through three of the four points $(0,0), (4,0), (-1,1), (4,2)$ is $\_\_\_\_$.

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 equation of a circle passing through three of the four points $( 0,0 ) , ( 4,0 ) , ( - 1,1 ) , ( 4,2 )$ is $\_\_\_\_$ .

16. Given an ellipse $C : \frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ ( $a > b > 0$ ), with upper vertex $A$, two foci $F _ { 1 }$ and $F _ { 2 }$, and eccentricity $\frac { 1 } { 2 }$. A line through $F _ { 1 }$ perpendicular to $A F _ { 2 }$ intersects $C$ at points $D$ and $E$, with $| D E | = 6$. The perimeter of $\triangle A D E$ is $\_\_\_\_$ .
IV. Solution Questions: This section contains 6 questions, for a total of 70 points. Solutions should include explanations, proofs, or calculation steps.

The eccentricity of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 \ (a > 0 , b > 0)$ is $\sqrt{5}$ . One of its asymptotes intersects the circle $(x - 2)^{2} + (y - 3)^{2} = 1$ at points $A , B$ , then $|AB| =$
A. $\frac{1}{5}$
B. $\frac{\sqrt{5}}{5}$
C. $\frac{2\sqrt{5}}{5}$
D. $\frac{4\sqrt{5}}{5}$

Circle $\odot O$ has radius 1. Line $PA$ is tangent to $\odot O$ at point $A$. Line $PB$ intersects $\odot O$ at points $B$ and $C$. $D$ is the midpoint of $BC$. If $| P O | = \sqrt { 2 }$, then the maximum value of $\overrightarrow { P A } \cdot \overrightarrow { P D }$ is
A. $\frac { 1 + \sqrt { 2 } } { 2 }$
B. $\frac { 1 + 2 \sqrt { 2 } } { 2 }$
C. $1 + \sqrt { 2 }$
D. $2 + \sqrt { 2 }$

Find the distance from the center of the circle $x ^ { 2 } + y ^ { 2 } - 2 x + 6 y = 0$ to the line $x - y + 2 = 0$

Given curve $C : x ^ { 2 } + y ^ { 2 } = 16 ( y > 0 )$, from any point $P$ on $C$, draw a perpendicular segment $P P ^ { \prime }$ to the $x$-axis, where $P ^ { \prime }$ is the foot of the perpendicular. The locus of the midpoint of segment $P P ^ { \prime }$ is
A. $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 4 } = 1 \quad ( y > 0 )$
B. $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 8 } = 1 \quad ( y > 0 )$
C. $\frac { y ^ { 2 } } { 16 } + \frac { x ^ { 2 } } { 4 } = 1 \quad ( y > 0 )$
D. $\frac { y ^ { 2 } } { 16 } + \frac { x ^ { 2 } } { 8 } = 1 \quad ( y > 0 )$

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. $(0, +\infty)$

Given the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ with eccentricity $\frac{\sqrt{2}}{2}$ and major axis length 4.
(1) Find the equation of $C$;
(2) A line $l$ passing through the point $(0, -2)$ intersects $C$ at points $A, B$. Let $O$ be the origin. If the area of $\triangle OAB$ is $\sqrt{2}$, find $|AB|$.

For $A = \left(\begin{array}{ll} a & b \\ c & d \end{array}\right)$ in $\mathcal{M}_2(\mathbb{R})$ and $(x,y)$ in $\mathbb{R}^2$, we denote by $\varphi_A(x,y)$ the determinant of the matrix $A(x,y) = \left(\begin{array}{cc} a-x & b-y \\ c+y & d-x \end{array}\right)$ and we consider $\mathcal{CP}_A$ the curve in $\mathbb{R}^2$ defined by the equation: $\varphi_A(x,y) = 0$.
Verify that $\mathcal{CP}_A$ is a circle (we agree that a circle can be reduced to a point); we will call $\mathcal{CP}_A$ the eigenvalue circle of $A$. Specify its center $C_A$ and its radius $r_A$.

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 $\mathbb{R} \times \{0\}$.

Exercise VII

Consider the points $A$ and $B$ with respective coordinates in an orthonormal coordinate system: $$A ( 2 ; 0 ) \text { and } \mathrm { B } ( 0 ; - 4 ) \text {. }$$ VII-A- An equation of the line $( A B )$ is $2 x - y - 4 = 0$. VII-B- An equation of the perpendicular bisector of segment $[ A B ]$ is $x + 2 y + 3 = 0$. VII-C- An equation of the circle with diameter $[ A B ]$ is $x ^ { 2 } + y ^ { 2 } - 2 x - 4 y = 0$. VII-D- The point with coordinates $( - 1 ; - 1 )$ belongs to the circle with diameter $[ A B ]$. VII-E- The line with equation $2 x - y + 1 = 0$ is tangent to the circle with diameter $[ A B ]$.
For each statement, indicate whether it is TRUE or FALSE.

Tangents are drawn to a given circle from a point on a given straight line, which does not meet the given circle. Prove that the locus of the mid-point of the chord joining the two points of contact of the tangents with the circle is a circle.

Let $A, B, C, D, E$ be the vertices of a regular pentagon inscribed in a circle of radius $r$. Let $F$ be the midpoint of side $AB$. Find the circumradius $AO$ in terms of the side length $x = AB$.

A circle is inscribed in a triangle with sides $8, 15, 17$ cms. The radius of the circle in cms is
(a) 3
(b) $22/7$
(c) 4
(d) None of the above.

Let $C _ { 1 } , C _ { 2 }$ and $C _ { 3 }$ be three circles lying in the same quadrant, each touching both the axes. Suppose also that $C _ { 1 }$ touches $C _ { 2 }$ and $C _ { 2 }$ touches $C _ { 3 }$. If the area of the smallest circle is 1 unit, then area of the largest circle is
(a) $\{ ( \sqrt{2} + 1 ) / ( \sqrt{2} - 1 ) \} ^ { 4 }$
(b) $( 1 + \sqrt{2} ) ^ { 2 }$
(c) $( 2 + \sqrt{2} ) ^ { 2 }$
(d) $2 ^ { 4 }$

Find the equation of the circle with $AB$ as diameter, where $A$ and $B$ are the intercepts of the line $2x + 3y = k$ on the coordinate axes.

Let $n$ be a positive integer. Consider a square $S$ of side $2n$ units. Divide $S$ into $4n^2$ unit squares by drawing $2n - 1$ horizontal and $2n - 1$ vertical lines one unit apart. A circle of diameter $2n - 1$ is drawn with its centre at the intersection of the two diagonals of the square $S$. How many of these unit squares contain a portion of the circumference of the circle?
(A) $4n - 2$
(B) $4n$
(C) $8n - 4$
(D) $8n - 2$

An isosceles triangle with base 6 cms. and base angles $30^\circ$ each is inscribed in a circle. A second circle touches the first circle and also touches the base of the triangle at its midpoint. If the second circle is situated outside the triangle, then its radius (in cms.) is
(A) $3\sqrt{3}/2$
(B) $\sqrt{3}/2$
(C) $\sqrt{3}$
(D) $4/\sqrt{3}$