(1) Find the analytical formula of parabola.
(2) It is known that AD = AB(D is on the line segment AC), and a moving point P moves from point A along the line segment AC at a speed of/kloc-0 per second; At the same time, another moving point Q moves from point B along BC line at a certain speed. After moving for t seconds, divide the line PQ vertically by BD to find the value of t;
(3) In the case of (2), is there a point m on the parabola axis of symmetry that minimizes the value of MQ+MC? If it exists, request the coordinates of point m; If it does not exist, please explain why.
(Note: Parabolic symmetry axis is)
(08 Analysis of 26 Questions in Putian, Fujian) (1) Solution 1: Let the analytical formula of parabola be y = a (x +3 )(x-4).
Because B (0 0,4) is on a parabola, 4 = a (0+3) (0-4) is solved to get a=-1/3.
So the parabolic analytical formula is
Solution 2: Let the analytical formula of parabola be,
According to the meaning of the question: c=4 and solve it.
So the analytical formula of parabola is
(2) connect DQ, in rt delta AOB,
So AD=AB= 5, AC=AD+CD=3+4 = 7, CD = AC-AD = 7-5 = 2.
Because BD vertically divides PQ, PD=QD, PQ⊥BD, so ∠PDB=∠QDB.
Because AD=AB, ∠ABD=∠ADB, ∠ABD=∠QDB, DQ ∠ AB.
So ∠CQD=∠CBA. ∠CDQ =∠ cab, so △CDQ∽△ cab.
that is
So AP = ad-DP = ad-dq = 5 -=–=,
So the value of t is
(3) There is a point m on the symmetry axis that minimizes the value of MQ+MC.
Reason: Because the symmetry axis of parabola is
Therefore, A (-3,0) and C (4 4,0) are symmetrical about a straight line.
If the intersection line connecting AQ is at point M, the value of MQ+MC is the smallest.
Q is QE⊥x axis, at point E, so ∠QED=∠BOA=900.
DQ‖AB，∠ BAO=∠QDE，△DQE ∽△ABO
that is
So QE=, Germany =, so OE = OD+ Germany =2+ =, so Q (,).
Let the analytical formula of straight line AQ be
Then the next step is
Therefore, the analytical formula of straight line AQ is simultaneous.
Therefore, m
Then: there is a point m on the axis of symmetry that minimizes the value of MQ+MC.
2(08 Gansu Baiyin and other 9 cities and 28 questions) (12 points) As shown in Figure 20, in the plane rectangular coordinate system, the quadrilateral OABC is rectangular, and the coordinate of point B is (4,3). The straight line M parallel to the diagonal AC starts from the origin O and moves at the speed of 1 unit length per second along the positive direction of the X axis. Both sides of the straight line M intersect with the right angle OABC at the point M respectively.
(1) The coordinates of point A are _ _ _ _ _ _ _, and the coordinates of point C are _ _ _ _ _ _ _ _ _;
(2) When t= seconds or seconds, MN = AC；;
(3) Let the area of △OMN be S, and find the functional relationship between S and T;
(4) Does the function S obtained in (3) have a maximum value? If yes, find the maximum value; If not, explain why.
(08 Analysis of 28 questions in 9 cities such as Baiyin, Gansu) The full score of this small question is 12.
Solution: (1) (4,0), (0,3); 2 points
(2) 2,6; 4 points
(3) When 0 < t ≤ 4, OM = t. 。
From △OMN∽△OAC
∴ In =, s = .6 points
When 4 < t < 8,
As shown in the figure, od = t, ∴ ad = t-4.
Method 1:
From △DAM∽△AOC, we can get AM=, ∴ BM = 6-.7 points.
From △BMN∽△BAC, BN= =8-t, you can get △ CN = t-4.8.
S= rectangular OABC area -Rt area △OAM-Rt area △MBN-Rt area △ NCO.
= 12- - (8-t)(6- )-
=. 10 point
Method 2:
It is easy to know that the quadrilateral ADNC is a parallelogram, ∴ CN=AD=t-4, bn = 8-t.7 points.
From △BMN∽△BAC, BM= =6-, you can get ∴ AM = .8 points.
The following is the same as method 1.
(4) There is a maximum value.
Method 1:
When 0 < t ≤ 4,
The opening of parabola S= is upward, and on the right side of symmetry axis t=0, s increases with the increase of t,
When t=4, the maximum value of s = 6; 1 1 min
When 4 < t < 8,
∵ The opening of parabola S= is downward, and its vertex is (4,6), ∴ s < 6.
To sum up, when t=4, the maximum value of S is 6. 12 points.
Method 2:
∫S =
∴ When 0 < t < 8, draw the functional relationship diagram between S and T, as shown in the figure. 1 1.
Obviously, when t=4, the maximum value of S is 6. 12 points.
Note: Only when the answer to question (3) is correct and the answer to question (4) is only "maximum" and there are no other steps, the score can be1; Otherwise, no points will be given.
3(08 Guangdong-Guangzhou 25 questions) (14 points) as shown in figure 1 1, in trapezoidal ABCD, AD‖BC, AB=AD=DC=2cm, BC=4cm, in isosceles △PQR, ∠ QPR. If the isosceles △PQR moves at a uniform speed 1cm/ s along the direction indicated by the arrow of the straight line L, the area of the overlapping part between the trapezoidal ABCD and the isosceles △PQR in t seconds is recorded as s square centimeters.
(1) When t=4, find the value of s.
(2) If the functional relationship between S and T is found, the maximum value of S is found.
(08 Analysis of 25 Questions in Guangzhou, Guangdong) (1) When t = 4, Q and B coincide, and P and D coincide.
Overlap =
4(08 Shenzhen, Guangdong, Question 22) As shown in Figure 9, in the plane rectangular coordinate system, the vertex of the image of the quadratic function is point D, which intersects with the Y axis at point C, and intersects with the X axis at points A and B. Point A is on the left side of the origin, and the coordinate of point B is (3,0).
OB=OC，tan∠ACO=。
(1) Find the expression of this quadratic function.
(2) The straight line passing through points C and D intersects the X axis at point E. Is there such a point F on this parabola, and the quadrilateral with points A, C, E and F as its vertices is a parallelogram? If it exists, request the coordinates of point f; If it does not exist, please explain why.
(3) If the straight line parallel to the X axis intersects the parabola at two points, M and N, and the circle with the diameter of MN is tangent to the X axis, find the length of the radius of the circle.
(4) As shown in figure 10, if point G(2, y) is a point on the parabola and point P is a moving point on the parabola below the straight line AG, when point P moves to what position, what is the maximum area of △APG? Find the coordinates of point P and the maximum area of △APG at this time.
(08 Analysis of 22 Questions in Shenzhen, Guangdong) (1) Method 1: From the known: C(0, -3), a (- 1, 0) … 1.
Substitute the coordinates of a, b and c.
Solution: 3 points.
So the expression of this quadratic function is
Method 2: From the known: C(0, -3), a (- 1, 0) ...............................1min.
Let this expression be
Substitute the coordinates of point C.
So the expression of this quadratic function is
(Note: The final result of the expression will not be deducted in any of the three forms)
(2) Method 1: exists, and the coordinates of point F are (2, -3)4 points.
Because: D( 1, -4) is easy to get, the analytical formula of linear CD is:
∴ The coordinate of point E is (-3,0) ............................ 4 points.
From the coordinates of a, c, e and f, AE = cf = 2, AE ‖ cf.
A quadrilateral with vertices a, c, e and f is a parallelogram.
∴ There is point F, and the coordinates are (2, -3). ...................................................................................................................................................
Method 2: D( 1, -4) is easy to obtain, so the analytical formula of linear CD is:
∴ The coordinate of point E is (-3,0) ............................ 4 points.
A quadrilateral with vertices A, C, E and F is a parallelogram.
∴ The coordinates of point F are (2, -3) or (-2, -3) or (-4, 3).
Only (2, -3) satisfies the parabolic expression test.
∴ There is point F, and the coordinates are (2, -3). ...................................................................................................................................................
(3) As shown in the figure, ① When the straight line MN is above the X axis, let the radius of the circle be r (r >; 0), then N(R+ 1, r),
Substituting into the expression of parabola, the solution is
② When the straight line MN is below the X axis, let the radius of the circle be r (r >; 0),
Then N(r+ 1, -r),
Substitute the parabola expression and you get ... 7 points.
The radius of a circle is 7 points or ........................
(4) When the Y axis intersects AG at point Q, the parallel line passing through point P,
G(2, -3) is easily obtained, and the straight line AG is ........................................................................................................................................................
Let P(x,), then Q(x, -x- 1), pq.
Nine points
When △APG has the largest area.
At this point, the coordinates of point p are, ...................................................................................................................................................................
5(08 Enshi, Hubei, 24 questions) (the full score of this big question is 12 points) as shown in figure 1 1. Put two isosceles right triangles ABC and AFG together on the same plane, where A is the common vertex and ∠ BAC = ∠ AGF = 90. ABC is fixed. AFG rotates around point A, and the intersections of AF, AG and BC are D and E respectively (point D and point B do not coincide, and point E does not coincide with point C). Let BE=m and CD = n.
(1) Please find two pairs of similar but unequal triangles in the diagram and choose one pair to prove it.
(2) Find the functional relationship between m and n, and write the range of independent variable n directly.
(3) same? The straight line on the hypotenuse BC of ABC is the X axis, and the straight line on the height of BC is the Y axis, thus establishing a plane rectangular coordinate system (as shown in figure 12). Find a point D on the side of BC so that BD=CE, find out the coordinates of point D, and verify BD +CE =DE through calculation.
(4) Whether the equivalence relation BD +CE =DE in (3) always holds in the process of rotation, if so, please prove it, if not, please explain the reasons.
(08 Analysis of 24 Questions in Enshi, Hubei Province) (Full score for this big question 12)
Solution: (1)? Abe? DAE？ Abe? DCA 1 min
∠∠BAE =∠BAD+45，∠CDA=∠BAD+45
∴∠BAE=∠CDA
∠ b =∠ c = 45。
∴? Abe? DCA 3 points
(2)∵? Abe? Defense Communications Agency
∴
According to the meaning of the question, CA=BA=
∴
M = 5 points
The range of independent variable n is 1
(3) BD=CE, BE=CD, that is, m = n.
∫m =
∴m=n=
∫OB = OC = BC = 1
∴OE=OD= - 1
∴ d (1-0) 7 points
∴bd=ob-od= 1-(- 1)= 2-= ce，DE=BC-2BD=2-2(2- )=2 -2
∫BD+CE = 2bd = 2(2-)= 12-8，DE =(2 -2) = 12-8
∴ BD+CE = DE 8 points
49 points.
Proof: as shown in the picture, will it? ACE rotates 90 clockwise around point a to? The position of ABH, CE=HB, AE=AH,
∠ ABH =∠ C = 45, and the rotation angle ∠ EAH = 90.
Connect HD, at? EAD and? Hade clock
AE = AH，∠HAD=∠EAH-∠FAG=45 =∠EAD，AD=AD。
∴? EAD？ have
∴DH=DE
And < hbd = < abh+< Abd = 90.
∴bd+ hemoglobin =DH
That is BD+CE = DE 12.
6(08 Jingmen, Hubei, 28 questions) (full score for this small question 12 points)
It is known that the vertex A of the parabola y=ax2+bx+c is on the X axis, and the intersection with the Y axis is b (0, 1), and b =-4ac.
(1) Find the analytical formula of parabola;
(2) Is there a point C on the parabola that makes the circle with diameter BC pass through the vertex A of the parabola? If there is no explanation; If it exists, find the coordinates of point C and the coordinates of the center point P of the circle at this time;
(3) According to the conclusion of (2), what is the relationship between the abscissa and ordinate of B, P and C?
Solution: (1) C = 1 When a parabola passes through B(0, 1).
And b=-4ac, vertex A(-, 0),
∴-= = 2c = 2。 ∴ A (2 2,0) ............................................................................. 2 points.
Substitute the coordinates of point A into the parabolic analytical formula, 4a+2b+ 1=0,
The solution of ∴ is a =, b =- 1.
Therefore, the analytical formula of parabola is y = x2-x+1................................................................................................ 4 minutes.
Another solution: C = 1, b2-4ac=0, b=-4ac, ∴ B =- 1 ..............................................................................................................
∴a=, so y = x-x+ 1 ..............................................................................................................................................
(2) Suppose that a point C satisfying the meaning of the question exists, and its coordinates are C(x, y).
Make CD⊥x axis on D, and connect AB and AC.
∫a On a circle with BC as its diameter, ∴∠ BAC = 90.
∴ △AOB∽△CDA。
∴OB？ CD=OA？ Advertising.
That is 1? Y=2(x-2), ∴ y = 2x-4 ............................... 6 points.
From the solution, X 1 = 10, X2 = 2.
Point C that satisfies the meaning of the question exists, and the coordinates are (10, 16), or (2,0). .................................................................................................................
∵P is the center of the circle and ∴P is the midpoint of BC.
When the coordinates of point C are (10, 16), take the OD midpoint P 1 and connect it with PP 1, then PP 1 is the center line of trapezoidal OBCD.
∴PP 1= (OB+CD)=。 ∫d( 10，0)，∴P 1 (5，0)，∴P (5，)。
When the coordinate of point C is (2,0), take the midpoint P2 of OA and connect PP2, then PP2 is the center line of △OAB.
∴PP2= OB=。 ∫a(2，0)，∴P2( 1,0)，∴P ( 1，)。
Therefore, the coordinates of point P are (5,) or (1,) ..................................................................................................... 10.
(3) Let the coordinates of b, p and c be B (x 1, y 1), P (x2, y2) and C (x3, y3), which is given by (2):
............................ 12.
7 (24 questions in Xianning, Hubei Province, 08) (this question (1) ~ (3) full score 12, (4) subtitle plus points)
As shown in figure 1, in the square ABCD, the coordinates of point A and point B are (0, 10) and (8,4) respectively, and point C is in the first quadrant. The moving point P is on the side of the square ABCD, and moves at a uniform speed from point A along A→B→C→D, while the moving point Q moves at a uniform speed on the X axis. When point p reaches point d, two points.
(1) When the point P moves on the edge AB, the function image of the abscissa (length unit) of the point Q with respect to the moving time t (seconds) is shown in Figure ②. Please write down the coordinates when point Q starts to move and the moving speed of point P;
(2) Find the coordinates of the side length and vertex c of the square;
(3) When t is at (1), the area of △OPQ is the largest, and the coordinates of point p at this time are obtained.
(1) Additional questions: (You can continue when you have time.
Answer the following questions and wish you success! )
If point P and point Q keep the original speed, the speed is not.
Change, when p points are evenly distributed along a → b → c → d.
Whether OP and PQ are equal when moving at high speed,
If possible, write down all qualified T's.
Value; If not, please explain why.
(08 Analysis of 24 Questions in Xianning, Hubei Province) Solution: (1) (1, 0)-1.
The moving speed of point P is 1 unit length per second. -Three minutes.
(2) If the passing point is the BF⊥y axis at the point and the ⊥ axis at the point, then = 8.
∴ .
In Rt△AFB, ...-5 points.
The intersection point is the axis at this point, and the extension line intersects with this point.
* ∴△abf≌△bch.
∴ .
∴ .
∴ The coordinates of point C are (14, 12). -Seven points.
(3) taking the crossing point P as the PM⊥y axis of point M and the PN⊥ axis of point N,
Then △APM∽△ABF.
∴ .。
∴ .∴ .
Let the area of △OPQ be (square unit)
∴ (0 ≤ 10)-10.
Note: If the range of independent variables is not indicated, no points will be deducted.
∵& lt； When ∴ is 0, the area of △OPQ is the largest. - 1 1.
At this point, the coordinates of p are (,). -12 points.
(4) When or and OP are equal to PQ. -14 points.
1 points plus one, no need to write the solution process.
8 (Question 26, Changsha, Hunan, 08) As shown in the figure, the hexagon ABCDEF is inscribed with a radius r (constant) ⊙O, where AD is the diameter and AB=CD=DE=FA.
(1) when ∠BAD=75? When, ask the length of BC ⌒;
(2) verification: BC ‖ AD ‖ Fe;
(3) Let AB=, find the functional relationship about the circumference L of hexagonal ABCDEF, and point out why L is the maximum.
(08 Analysis of Question 26 in Changsha, Hunan) (1) Connect OB and OC from ∠BAD=75? , OA=OB know ∠AOB=30? , (1 min)
∵AB=CD,∴∠COD=∠AOB=30？ ,∴∠BOC= 120？ , (2 points)
Therefore, the length of BC⌒ is. (3 points)
(2) link BD, ab = cd, ∴∠ADB=∠CBD, ∴BC‖AD, (5 points)
Similarly, EF‖AD, so BC ‖ ad ‖ Fe. (6 points)
(3) If the intersection point b is BM⊥AD in m, it can be seen from (2) that the quadrilateral ABCD is an isosceles trapezoid, so BC = AD-2am = 2r-2am. (7 points)
∫ad is the diameter, ∴∠ABD=90? , easy to get △BAM∽△DAB
∴AM= =, ∴BC=2r-, and the same EF=2r- (8 points)
∴L=4x+2(2r- )= =, where 0 < x < (9 points)
When x=r, l gets the maximum value 6r. (10).
9 (24 questions in Yiyang, Hunan in 2008) (this question 12 points) We call a closed figure composed of a semicircle and a part of a parabola "egg circle". If a straight line has only one intersection with the "egg circle", then this straight line is called the tangent of the "egg circle".
As shown in figure 12, points A, B, C and D are the intersections of the "egg circle" and the coordinate axis respectively. It is known that the coordinate of point D is (0, -3), AB is the diameter of semicircle, the m coordinate of the center of semicircle is (1, 0), and the radius of semicircle is 2.
(1) Please find out the analytical formula of the parabolic part of the "egg circle" and write down the range of the independent variable;
(2) Can we find the analytical formula of the tangent of the "egg circle" passing through point C? Give it a try;
(3) Use your head and think about it. I believe you can find the analytical formula of the tangent of the egg circle passing through point D.
(08 Analysis of Question 24 in Yiyang, Hunan) (This question 12 points) Solution: (1) Solution 1: According to the meaning of the question, the following can be obtained: A(- 1 0), B (3 3,0);
Let the analytical expression of parabola be (a≠0)
The other point D(0, -3) is on the parabola, and ∴a(0+ 1)(0-3)=-3, and the solution is: a= 1.
Y = x2-2x-3 3 points
Independent variable range:-1≤x≤3 4 points.
Solution 2: Let the analytical expression of parabola be (a≠0).
According to the meaning of the question, A(- 1 0), B (3,0) and D (0 0,3) are all on a parabola.
∴, solution:
Y = x2-2x-3 3 points
Independent variable range:-1≤x≤3 4 points.
(2) Let the tangent CE passing through the "egg circle" at point C intersect with the X axis and connect CM at point E,
In Rt△MOC, ∫om = 1, CM=2, ∴∠ CMO = 60, OC =
In Rt△MCE, oc = 2, ∠ CMO = 60, ∴ME=4.
∴ The coordinates of point C and point E are (0,) and (-3, 0) respectively.
The analytical formula of tangent CE is 8 points.
(3) Let the intersection point D(0, -3), and the analytical formula of the tangent of the "egg circle" is: y=kx-3(k≠0) 9 points.
As can be seen from the meaning of the question, the equations have only one set of solutions.
That is, there are two equal real roots, ∴k=-2 1 1 point.
∴ Analytical formula of the tangent of the "egg circle" passing through point D =-2x-3 12 point.
10 (Nanjing, Jiangsu Province, 28 questions in 2008) (10 minute) An express train goes from A to B, and a local train goes from B to A, and both trains leave at the same time. Let the local train travel time be, and the distance between two cars be. The dotted line in the figure shows the functional relationship with.
According to the pictures, make the following inquiries:
Information reading
(1) the distance between a and b is km;
(2) Please explain the actual meaning of the dots in the picture;
Image understanding
(3) Find the speed of the local train and the express train;
(4) Find the relationship between the line segment and the function it represents, and write the range of independent variables;
problem solving
(5) If the second express also goes from A to B, the speed is the same as that of the first express. Thirty minutes after the first express train meets the local train, the second express train meets the local train. How many hours does it leave after the first express?
(08 analysis of 28 questions in Nanjing, Jiangsu) 28. (This question 10)
Solution: (1) 900; 1 point
(2) The practical significance of the midpoint in the figure is that when the local train runs for 4 hours, the local train meets the express train. 2 points.
(3) According to the image, the traveling distance of the local train 12h is 900km.
So the speed of the local train is; 3 points
When the local train runs for 4 hours, the local train meets the express train, and the sum of the distances traveled by the two trains is 900km, so the sum of the speeds of the local train and the express train is 150 km/h.4 minutes.
(4) According to the meaning of the question, the express train travels 900km to the second place, so the express train travels to the second place. At this time, the distance between the two cars is, so the coordinates of the point are.
Let the functional relationship between the sum represented by the line segment be, substitution.
solve
Therefore, the functional relationship between the line segment and is .6 points.
The range of independent variables is .7 points.
(5) After 30 minutes, the local train meets the first express train and the second express train. At this point, the running time of the local train is 4.5h 。
Substitute and get.
At this time, the distance between the local train and the first express train is equal to the distance between the two express trains, which is 1 12.5km, so the interval between the two express trains is, that is, the second express train leaves 0.75 h later than the first express train10 minute.