8. Circle C has its center on the line $2 x - y - 7 = 0$ and intersects the y-axis at two points $A ( 0 , - 4 )$ and $B ( 0 , - 2 )$. The equation of circle C is $\_\_\_\_$.

The equation of a circle with center $(1,1)$ and passing through the origin is

16. As shown in the figure, circle C is tangent to the x-axis at point $T ( 1,0 )$, and intersects the positive y-axis at two points $\mathrm { A } , \mathrm { B }$ (B is above A), with $| A B | = 2$.
(1) The standard equation of circle C is $\_\_\_\_$.
(2) The x-intercept of the tangent line to circle C at point B is $\_\_\_\_$. [Figure]

18. (This problem is worth 16 points) As shown in the figure, in the rectangular coordinate system xOy, given that the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ has eccentricity $\frac { \sqrt { 2 } } { 2 }$, and the distance from the right focus F to the left directrix l is 3. [Figure]
(1) Find the standard equation of the ellipse;
(2) A line through F intersects the ellipse at points $\mathrm { A } , \mathrm { B }$. The perpendicular bisector of segment AB intersects the line l and AB at points $\mathrm { P } , \mathrm { C }$ respectively. If $\mathrm { PC } = 2 \mathrm { AB }$, find the equation of line AB.

15. For the ellipse $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ , the right focus $\mathrm { F } ( c , 0 )$ is symmetric to point Q with respect to the line $y = \frac { b } { c } x$ , and Q lies on the ellipse. Then the eccentricity of the ellipse is $\_\_\_\_$. III. Solution Questions (This section contains 5 questions, 74 points total. Solutions should include explanations, proofs, or calculation steps.)

The parabola $C : x ^ { 2 } = - 2 p y$ passes through the point $( 2 , - 1 )$. (I) Find the equation of parabola $C$ and the equation of its directrix; (II) Let $O$ be the origin. A line $l$ with non-zero slope passes through the focus of parabola $C$ and intersects parabola $C$ at two points $M , N$. The line $y = - 1$ intersects lines $O M$ and $O N$ at points $A$ and $B$ respectively. Prove that the circle with $A B$ as diameter passes through two fixed points on the $y$-axis.

10. Given that the foci of ellipse $C$ are $F _ { 1 } ( - 1,0 ) , F _ { 2 } ( 1,0 )$ , and a line through $F _ { 2 }$ intersects $C$ at points $A , B$ . If $\left| A F _ { 2 } \right| = 2 \left| F _ { 2 } B \right|$ and $| A B | = \left| B F _ { 1 } \right|$ , then the equation of $C$ is
A. $\frac { x ^ { 2 } } { 2 } + y ^ { 2 } = 1$
B. $\frac { x ^ { 2 } } { 3 } + \frac { y ^ { 2 } } { 2 } = 1$
C. $\frac { x ^ { 2 } } { 4 } + \frac { y ^ { 2 } } { 3 } = 1$
D. $\frac { x ^ { 2 } } { 5 } + \frac { y ^ { 2 } } { 4 } = 1$

9. If the focus of the parabola $y ^ { 2 } = 2 p x ( p > 0 )$ is a focus of the ellipse $\frac { x ^ { 2 } } { 3 p } + \frac { y ^ { 2 } } { p } = 1$, then $p =$
A. 2
B. 3
C. 4
D. 8

8. If the focus of the parabola $y ^ { 2 } = 2 p x ( p > 0 )$ is a focus of the ellipse $\frac { x ^ { 2 } } { 3 p } + \frac { y ^ { 2 } } { p } = 1$, then $p =$
A. $2$
B. $3$
C. $4$
D. $8$

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 $\_\_\_\_$ .

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

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 $r_A$.

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.

136- The center of a circle is on the first-quadrant angle bisector. If this circle passes through point $A(6,3)$ and is tangent to the line $y = 2x$, what is its radius?
(1) $\sqrt{5}$ (2) $\sqrt{6}$ (3) $2\sqrt{2}$ (4) $\sqrt{10}$

133- The smallest circle passing through the two points $A(2,5)$ and $B(-4,1)$ intersects the $x$-axis at what length?

• [(1)] $1\ ,\ -3$
• [(2)] $5\ ,\ -3$
• [(3)] $2\ ,\ -1$
• [(4)] $3\ ,\ -2$

134- Among the circles passing through the point $A(-4\ ,\ 1)$ and tangent to the lines $4x + 3y = 0$ and the $y$-axis, the one with the largest radius is:

• [(1)] $\dfrac{5}{3}$
• [(2)] $\dfrac{17}{9}$
• [(3)] $\dfrac{7}{3}$
• [(4)] $\dfrac{22}{9}$

140- For every $m$, the equation $y = 6$, $(m+1)x + (m-2)y = 6$ is the equation of a chord of circle $C$. If point $A(-1,1)$ lies on circle $C$, the circumference of circle $C$ is which?
\[ \text{(1)}\ 2\sqrt{3}\pi \qquad \text{(2)}\ 2\pi \qquad \text{(3)}\ 3\pi \qquad \text{(4)}\ 2\sqrt{7}\pi \]
%% Page 9

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.

12. A circle touches the line $2 x + 3 y + 1 = 0$ at the point $( 1 , - 1 )$ and is orthogonal to the circle which has the line segment having end points $( 0 , - 1 )$ and $( - 2,3 )$ as the diameter.
Sol. Let the circle with tangent $2 \mathrm { x } + 3 \mathrm { y } + 1 = 0$ at $( 1 , - 1 )$ be $( x - 1 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + \lambda ( 2 x + 3 y + 1 ) = 0$ or $x ^ { 2 } + y ^ { 2 } + x ( 2 \lambda - 2 ) + y ( 3 \lambda + 2 ) + 2 + \lambda = 0$. It is orthogonal to $\mathrm { x } ( \mathrm { x } + 2 ) + ( \mathrm { y } + 1 ) ( \mathrm { y } - 3 ) = 0$ Or $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } + 2 \mathrm { x } - 2 \mathrm { y } - 3 = 0$ so that $\frac { 2 ( 2 \lambda - 2 ) } { 2 } \cdot \left( \frac { 2 } { 2 } \right) + \frac { 2 ( 3 \lambda + 2 ) } { 2 } \left( \frac { - 2 } { 2 } \right) = 2 + \lambda - 3 \Rightarrow \lambda = - \frac { 3 } { 2 }$. Hence the required circle is $2 x ^ { 2 } + 2 y ^ { 2 } - 10 x - 5 y + 1 = 0$.

12. A circle touches the line $2 x + 3 y + 1 = 0$ at the point $( 1 , - 1 )$ and is orthogonal to the circle which has the line segment having end points $( 0 , - 1 )$ and $( - 2,3 )$ as the diameter.
Sol. Let the circle with tangent $2 \mathrm { x } + 3 \mathrm { y } + 1 = 0$ at $( 1 , - 1 )$ be $( x - 1 ) ^ { 2 } + ( y + 1 ) ^ { 2 } + \lambda ( 2 x + 3 y + 1 ) = 0$ or $x ^ { 2 } + y ^ { 2 } + x ( 2 \lambda - 2 ) + y ( 3 \lambda + 2 ) + 2 + \lambda = 0$. It is orthogonal to $\mathrm { x } ( \mathrm { x } + 2 ) + ( \mathrm { y } + 1 ) ( \mathrm { y } - 3 ) = 0$ Or $\mathrm { x } ^ { 2 } + \mathrm { y } ^ { 2 } + 2 \mathrm { x } - 2 \mathrm { y } - 3 = 0$ so that $\frac { 2 ( 2 \lambda - 2 ) } { 2 } \cdot \left( \frac { 2 } { 2 } \right) + \frac { 2 ( 3 \lambda + 2 ) } { 2 } \left( \frac { - 2 } { 2 } \right) = 2 + \lambda - 3 \Rightarrow \lambda = - \frac { 3 } { 2 }$. Hence the required circle is $2 x ^ { 2 } + 2 y ^ { 2 } - 10 x - 5 y + 1 = 0$.

A circle $C$ of radius 1 is inscribed in an equilateral triangle $P Q R$. The points of contact of $C$ with the sides $P Q , Q R , R P$ are $D , E , F$, respectively. The line $P Q$ is given by the equation $\sqrt { 3 } x + y - 6 = 0$ and the point $D$ is $\left( \frac { 3 \sqrt { 3 } } { 2 } , \frac { 3 } { 2 } \right)$. Further, it is given that the origin and the centre of $C$ are on the same side of the line $P Q$.
The equation of circle $C$ is
(A) $\quad ( x - 2 \sqrt { 3 } ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$
(B) $( x - 2 \sqrt { 3 } ) ^ { 2 } + \left( y + \frac { 1 } { 2 } \right) ^ { 2 } = 1$
(C) $\quad ( x - \sqrt { 3 } ) ^ { 2 } + ( y + 1 ) ^ { 2 } = 1$
(D) $( x - \sqrt { 3 } ) ^ { 2 } + ( y - 1 ) ^ { 2 } = 1$

Tangents drawn from the point $P ( 1,8 )$ to the circle
$$x ^ { 2 } + y ^ { 2 } - 6 x - 4 y - 11 = 0$$
touch the circle at the points $A$ and $B$. The equation of the circumcircle of the triangle $P A B$ is
(A) $x ^ { 2 } + y ^ { 2 } + 4 x - 6 y + 19 = 0$
(B) $x ^ { 2 } + y ^ { 2 } - 4 x - 10 y + 19 = 0$
(C) $x ^ { 2 } + y ^ { 2 } - 2 x + 6 y - 29 = 0$
(D) $x ^ { 2 } + y ^ { 2 } - 6 x - 4 y + 19 = 0$

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 the circle with AB as its diameter is
A) $x ^ { 2 } + y ^ { 2 } - 12 x + 24 = 0$
B) $x ^ { 2 } + y ^ { 2 } + 12 x + 24 = 0$
C) $x ^ { 2 } + y ^ { 2 } + 24 x - 12 = 0$
D) $x ^ { 2 } + y ^ { 2 } - 24 x - 12 = 0$

The circle passing through the point $( - 1,0 )$ and touching the $y$-axis at $( 0,2 )$ also passes through the point
(A) $\left( - \frac { 3 } { 2 } , 0 \right)$
(B) $\left( - \frac { 5 } { 2 } , 2 \right)$
(C) $\left( - \frac { 3 } { 2 } , \frac { 5 } { 2 } \right)$
(D) $( - 4,0 )$