8. A moving point $P$ has equal distance to point $F ( 2,0 )$ and to the line $x + 2 = 0$ . Then the locus equation of point $P$ is $\_\_\_\_$.

23. (Total Score: 18 points) Subproblem 1: 4 points, Subproblem 2: 6 points, Subproblem 3: 8 points.
Given that the equation of ellipse $\Gamma$ is $\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1 ( a > b > 0 )$ , with $A ( 0 , b ) , B ( 0 , - b )$ and $Q ( a , 0 )$ being three vertices of $\Gamma$.
(1) If point $M$ satisfies $\overrightarrow { A M } = \frac { 1 } { 2 } ( \overrightarrow { A Q } + \overrightarrow { A B } )$ , find the coordinates of point $M$;
(2) Let line $l _ { 1 } : y = k _ { 1 } x + p$ intersect ellipse $\Gamma$ at points $C , D$ and intersect line $l _ { 2 } : y = k _ { 2 } x$ at point $E$ . If $k _ { 1 } \cdot k _ { 2 } = - \frac { b ^ { 2 } } { a ^ { 2 } }$, prove that $E$ is the midpoint of $C D$;
(3) Let point $P$ be inside ellipse $\Gamma$ and not on the $x$-axis. How should one construct a line $l$ passing through the midpoint $F$ of $P Q$ such that the two intersection points $P _ { 1 } , P _ { 2 }$ of $l$ with ellipse $\Gamma$ satisfy $\overrightarrow { P P _ { 1 } } + \overrightarrow { P P _ { 2 } } = \overrightarrow { P Q }$ ? Let $a = 10 , b = 5$ , and the coordinates of point $P$ are $( - 8 , - 1 )$ . If points $P _ { 1 } , P _ { 2 }$ on ellipse $\Gamma$ satisfy $\overrightarrow { P P _ { 1 } } + \overrightarrow { P P _ { 2 } } = \overrightarrow { P Q }$ , find the coordinates of points $P _ { 1 } , P _ { 2 }$ .

2010 National College Entrance Examination Mathematics (Science) Shanghai Test

2010-6-7 Class $\_\_\_\_$ , Student ID $\_\_\_\_$ , Name $\_\_\_\_$ I. Fill in the Blanks (Total Score: 56 points, 4 points each)
1. The solution set of the inequality $\frac { 2 - x } { x + 4 } > 0$ is $\_\_\_\_$.
2. If the complex number $z = 1 - 2 i$ ($i$ is the imaginary unit), then $z \cdot \bar { z } + z =$ $\_\_\_\_$.
3. A moving From the system of equations $\left\{ \begin{array} { l } y = k _ { 1 } x + p \\ y = k _ { 2 } x \end{array} \right.$ , eliminating $y$ gives the equation $\left( k _ { 2 } - k _ { 1 } \right) x = p$ , Since $k _ { 2 } = - \frac { b ^ { 2 } } { a ^ { 2 } k _ { 1 } }$ , we have $\left\{ \begin{array} { l } x = \frac { p } { k _ { 2 } - k _ { 1 } } = - \frac { a ^ { 2 } k _ { 1 } p } { a ^ { 2 } k _ { 1 } ^ { 2 } + b ^ { 2 } } = x _ { 0 } \\ y = k _ { 2 } x = \frac { b ^ { 2 } p } { a ^ { 2 } k _ { 1 } ^ { 2 } + b ^ { 2 } } = y _ { 0 } \end{array} \right.$ , Therefore $E$ is the midpoint of $C D$ ;
(3) Since point $P$ is inside the ellipse $\Gamma$ and not on the $x$-axis, point $F$ is inside the ellipse $\Gamma$ . We can find the slope $k _ { 2 }$ of line $O F$ . From $\overrightarrow { P P _ { 1 } } + \overrightarrow { P P _ { 2 } } = \overrightarrow { P Q }$ we know that $F$ is the midpoint of $P _ { 1 } P _ { 2 }$ . According to (2), we can obtain the slope of line $l$ as $k _ { 1 } = - \frac { b ^ { 2 } } { a ^ { 2 } k _ { 2 } }$ , and thus obtain the equation of line $l$ . $F \left( 1 , - \frac { 1 } { 2 } \right)$ , the slope of line $O F$ is $k _ { 2 } = - \frac { 1 } { 2 }$ , the slope of line $l$ is $k _ { 1 } = - \frac { b ^ { 2 } } { a ^ { 2 } k _ { 2 } } = \frac { 1 } { 2 }$ , Solving the system of equations $\left\{ \begin{array} { l } y = \frac { 1 } { 2 } x - 1 \\ \frac { x ^ { 2 } } { 100 } + \frac { y ^ { 2 } } { 25 } = 1 \end{array} \right.$ , eliminating $y$ : $x ^ { 2 } - 2 x - 48 = 0$ , we obtain $P _ { 1 } ( - 6 , - 4 ) , P _ { 2 } ( 8,3 )$ .

Reference Answers for Science

I. Fill in the Blanks

$1 . ( - 4,2 )$ ;
2. $6 - 2i$ ;
3. $y ^ { 2 } = 8 x$ ;
4. $0$ ;
5. $3$ ; 6. $8.2$ ;

7. $S \leftarrow S + a$ ;

$8 . ( 0 , - 2 )$ ; 9. $\frac { 7 } { 26 }$ ; 10. $45$ ; 11. $1$ ; 12. $\frac { 8 \sqrt { 2 } } { 3 }$ ; 13. $4 a b = 1$ ; 14. $36$ .

II. Multiple Choice

15. A;

22. (Total: 16 points; Part 1: 4 points; Part 2: 6 points; Part 3: 6 points) Given the ellipse $C: \frac{x^2}{m^2} + y^2 = 1$ (constant $m > 1$), $P$ is a moving point on curve $C$, $M$ is the right vertex of curve $C$, and the fixed point $A$ has coordinates $(2, 0)$
(1) If $M$ coincides with $A$, find the coordinates of the foci of curve $C$;
(2) If $m = 3$, find the maximum and minimum values of $|PA|$;
(3) If the minimum value of $|PA|$ is $|MA|$, find the range of the real number $m$.

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

As shown in the figure, in circle O, M and N are trisection points of chord AB. Chords CD and CE pass through points M and N respectively. If $\mathrm{CM} = 2$, $\mathrm{MD} = 4$, $\mathrm{CN} = 3$, then the length of segment NE is
(A) $\frac{8}{3}$
(B) 3
(C) $\frac{10}{3}$
(D) $\frac{5}{2}$

5. As shown in the figure, let $F$ be the focus of the parabola $y ^ { 2 } = 4 x$. A line not passing through the focus contains three distinct points $A , B , C$, where points $A , B$ are on the parabola and point $C$ is on the $y$-axis. Then the ratio of the areas of $\triangle BCF$ and $\triangle ACF$ is
A. $\frac { | B F | - 1 } { | A F | - 1 }$
B. $\frac { | B F | ^ { 2 } - 1 } { | A F | ^ { 2 } - 1 }$
C. $\frac { | B F | + 1 } { | A F | + 1 }$
D. $\frac { | B F | ^ { 2 } + 1 } { | A F | ^ { 2 } + 1 }$

6. If one asymptote of the hyperbola $\frac { x ^ { 2 } } { a ^ { 2 } } - \frac { y ^ { 2 } } { b ^ { 2 } } = 1$ passes through the point $( 3 , - 4 )$, then the eccentricity of this hyperbola is
A. $\frac { \sqrt { 7 } } { 3 }$
B. $\frac { 5 } { 4 }$
C. $\frac { 4 } { 3 }$
D. $\frac { 5 } { 3 }$

6. As shown in the figure, in circle $O$, $M, N$ are trisection points of chord $AB$. Chords $CD, CE$ pass through points $M, N$ respectively. If $CM = 3$, then the length of segment $NE$ is [Figure]
(A) $\frac { 8 } { 3 }$
(B) 3
(C) $\frac { 10 } { 3 }$
(D) $\frac { 5 } { 2 }$

7. Given three points $A ( 1,0 ) , B ( 0 , \sqrt { 3 } ) , C ( 2 , \sqrt { 3 } )$, the distance from the circumcenter of $\triangle A B C$ [Figure]
to the origin is
A. $\frac { 5 } { 3 }$
B. $\frac { \sqrt { 21 } } { 3 }$
C. $\frac { 2 \sqrt { 5 } } { 3 }$
D. $\frac { 4 } { 3 }$

The circle passing through three points $A ( 1,3 ) , B ( 4,2 ) , C ( 1,7 )$ intersects the $y$-axis at points $\mathrm { M }$ and $\mathrm { N }$. Then $| M N | =$
(A) $2 \sqrt { 6 }$
(B) $8$
(C) $4 \sqrt { 6 }$
(D) $10$

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 $12$

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. $2 \sqrt { 10 }$

10. In the rectangular coordinate system $x O y$, among all circles with center at point $( 1,0 )$ and tangent to the line $m x - y - 2 m - 1 = 0 ( m \in R )$, the standard equation of the circle with the largest radius is $\_\_\_\_$.

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 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 $\_\_\_\_$ .

12. In the rectangular coordinate system xOyz, with the coordinate origin as the pole and the positive x-axis as the polar axis, if the polar equation of curve C is $\rho = 3 \sin \theta$, then the rectangular coordinate equation of curve C is $\_\_\_\_$

12. In the rectangular coordinate system $x O y$, let $P$ be a moving point on the right branch of the hyperbola $x ^ { 2 } - y ^ { 2 } = 1$. If the distance from point $P$ to the line $x - y + 1 = 0$ is always greater than or equal to c, then the maximum value of the real number c is $\_\_\_\_$.

13. If the line $3 x - 4 y + 5 = 0$ intersects the circle $x ^ { 2 } + y ^ { 2 } = r ^ { 2 } \quad ( r > 0 )$ at points $A$ and $B$, and $\angle A O B = 120 ^ { \circ }$ (where O is the coordinate origin), then $r =$ $\_\_\_\_$.

14. As shown in question (14), chords $\mathrm { AB }$ and $\mathrm { CD }$ of circle $O$ intersect at point $E$. A tangent line to circle $O$ is drawn through point $A$ and intersects the extension of $DC$ at point $P$. If $P A = 6 , A E = 9 , P C = 3 , C E : E D = 2 : 1$, then $B E = $ $\_\_\_\_$ . [Figure]

14. 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 $A$ and $B$ (with $B$ above $A$), and $|AB| = 2$. (I) The standard equation of circle $C$ is $\_\_\_\_$ ; (II) A line is drawn through point $A$ intersecting circle $O: x^2 + y^2 = 1$ at points $M$ and $N$. Consider the following three conclusions:
(1) $\frac{|NA|}{|NB|} = \frac{|MA|}{|MB|}$ ;
(2) $\frac{|NB|}{|NA|} - \frac{|MA|}{|MB|} = 2$ ;
(3) $\frac{|NB|}{|NA|} + \frac{|MA|}{|MB|} = 2\sqrt{2}$ .
The correct conclusion(s) is/are $\_\_\_\_$ . (Write the numbers of all correct conclusions)
(B) Optional Questions (Choose one of questions 15 and 16 to answer. First fill in the box after the question number you choose on the answer sheet with a 2B pencil. If you choose both, only question 15 will be graded.)

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} = $ $\_\_\_\_$ .

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.)

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]

16. (This question is worth 12 points) This question has three optional parts I, II, and III. Please select any two to answer and write your solutions in the corresponding answer areas on the answer sheet. If you answer all three, only the first two will be graded. I (This question is worth 6 points) Elective 4-1: Geometric Proof As shown in Figure 5, in circle O, two chords AB and CD intersect at point E, with midpoints M and N respectively. The line MO intersects line CD at point F. Prove: (I) $\angle \mathrm { MEN } + \angle \mathrm { NOM } = 180 ^ { \circ }$; (II) $\mathrm { FE } \cdot \mathrm { FN } = \mathrm { FM } \cdot \mathrm { FO }$
[Figure]
Figure 5
II. (This question is worth 6 points) Elective 4-4: Coordinate Systems and Parametric Equations Given the line $l: \left\{ \begin{array} { l } x = 5 + \frac { \sqrt { 3 } } { 2 } t \\ y = \sqrt { 3 } + \frac { 1 } { 2 } t \end{array} \right.$ (where t is the parameter). With the origin as the pole and the positive x-axis as the polar axis, the polar equation of curve C is $\rho = 2 \cos \theta$
(i) Convert the polar equation of curve C to rectangular coordinates; (II) Let the rectangular coordinates of point M be $( 5 , \sqrt { 3 } )$. The line $l$ intersects curve C at points $A$ and $B$. Find the value of $| M A | \cdot | M B |$ III. (This question is worth 6 points) Elective 4-5: Inequalities Let $\mathrm { a } > 0$, $\mathrm { b } > 0$, and $\mathrm { a } + \mathrm { b } = \frac { 1 } { a } + \frac { 1 } { b }$. Prove
(i) $\mathrm { a } + \mathrm { b } \geqslant 2$;
(ii) $\mathrm { a } ^ { 2 } + \mathrm { a } < 2$ and $\mathrm { b } ^ { 2 } + \mathrm { b } < 2$ cannot both be true.