8 1.(08 Maoming, Guangdong, 25 questions) (full mark for this question 10)
As shown in the figure, in the plane rectangular coordinate system, parabola =-++passes through three points, A(0, -4), B (0) and C (0), and -= 5.
(1) sum value; (4 points)
(2) Find a point D on the parabola so that the quadrilateral BDCE is a diamond with BC as the diagonal; (3 points)
(3) Is there a point P on the parabola that makes the quadrilateral BPOH a diamond with OB as the diagonal? If it exists, find the coordinates of point P and judge whether this diamond is a square. If it does not exist, please explain why. (3 points)
Solution:
(08 Analysis of Question 25 in Maoming, Guangdong) Solution: (1) Solution 1:
∵ parabola =-++passes through point A (0, -4),
=-4 ... 1 point
From the meaning of the problem, we can see that it is the two roots of the equation -++= 0,
+=, =-= 6 2 points
By the known (-) =25
(-) = (+)-4 =-24.
∴ -24=25
The solution is = 3 points.
When =, the intersection of the parabola and the axis is on the positive semi-axis of the axis, which is irrelevant and discarded.
=-0.4 points
Solution 2:∫ is the two roots of the equation -++c = 0.
That is, the two roots of equation 2-3+ 12 = 0.
=, 2 points
∴ - = =5,
The solution is = 3 points.
(The following is the same as Scheme 1. )
(2)∵ Quadrilateral BDCE is a diamond with BC as the diagonal. According to the nature of diamond, point D must be on the axis of symmetry of parabola, and there are five points.
And ∵ =-4 =-(+)+6 points.
The vertex (-,) of the parabola is point D.7.
(3)∵ Quadrilateral BPOH is a rhombus with OB as the diagonal, and the coordinate of point B is (-6,0).
According to the nature of the diamond, point P must be a straight line =-3 and.
The intersection of parabolas =-4, 8 points.
∴ When =-3, =-× (-3)-× (-3)-4 = 4,
There is a point p (-3,4) on the parabola, which makes the quadrilateral BPOH a diamond. Nine points
The quadrilateral BPOH can't be a square, because if the quadrilateral BPOH is a square, then the coordinate of the point P can only be (-3,3), but this point is not on the parabola. 10 point.
82. (25 questions in Zhaoqing, Guangdong Province in 2008) (Full score for this small question 10)
It is known that points A(a,), B(2a, y) and C(3a, y) are all on a parabola.
(1) Find the coordinates of the intersection of parabola and X axis;
(2) When a= 1, find the area of △ABC;
(3) Is there an equation that contains y, y and has nothing to do with A? If it exists, try to give one and prove it; If it does not exist, explain why.
(Analysis of 25 Questions in Zhaoqing, Guangdong, 08) (Full score for this small question 10)
Solution: (1) from 5 =0, (1)
Get,. (2 points)
∴ The coordinates of the intersection of parabola and X axis are (0,0), (0). (3 points)
(2) When a= 1, get A (1, 17), B (2 2,44), C (3 3,81), (4 points).
Points a, b and c are perpendicular to the x axis, and the vertical feet are d, e and f respectively, so there is
= s-(5 points)
=-(6 points)
=5 (unit area) (7 points)
Give an example. (8 points)
Actually, = 45a2+36a.
3()= 3[5×(2a)2+ 12×2a-(5 a2+ 12a)]= 45 a2+36a。 (9 points)
. ∴.( 10/0)
83.(08 Shenyang, Liaoning, 26 questions) (this question 14 points) 26. As shown in the figure, in the plane rectangular coordinate system, the side of the rectangle is on the negative semi-axis of the shaft, and the side is on the positive semi-axis of the shaft. After rotating clockwise around the point, the rectangle is obtained. The corresponding point of a point is a point, the corresponding point of a point is a point, the corresponding point of a point is a point, and a parabola passes through the point.
(1) Judge whether the point is on the axis and explain the reason;
(2) Find the function expression of parabola;
(3) Whether there is a point above the axis, so that the area of a parallelogram with this point as the vertex is twice that of a rectangle, and the point is on a parabola. If yes, find the coordinates of points and points; If it does not exist, please explain why.
(08 Shenyang, Liaoning 26 problem analysis) Solution: (1) point on the axis 1 point.
The reason for this is the following：
Connect, as shown, in,,,
,
According to the meaning of the question:
A point on the axis, a point on the axis. 3 points
(2) The intersection point is the axis of the point.
,
In,,
The point is in the first quadrant,
The coordinates of this point are 5 points.
According to (1), the point is on the positive semi-axis of the shaft.
The coordinates of this point are
The coordinates of this point are 6 points.
Parabolic passing point,
From the meaning of the question, substitute, into.
solve
The parabolic expression is: 9 points.
(3) There is a qualified score, and the score is. 10.
The reasons are as follows: the area of the rectangle
The area of a parallelogram with vertices is.
According to the meaning of the question, one side of this parallelogram,
and
The height of the side is 2 1 1.
The coordinates of the points set according to the meaning of the question are as follows
The point is on a parabola.
Solve,
,
A quadrilateral with vertices is a parallelogram,
, ,
When the coordinates of a point are,
The coordinates of the points are:
When the coordinates of a point are,
The coordinates of the points are. 14 points respectively.
84.(08 Liaoning 12 city 26 questions) (this question 14 points) 26. As shown in figure 16, in the plane rectangular coordinate system, the straight line intersects with the axis at one point, intersects with the axis at one point, and the parabola passes through three points.
(1) Find the analytical formula and vertex coordinates of a three-point parabola;
(2) Whether there is a point on the parabola, make it into a right triangle, and if there is, write the point coordinates directly; If it does not exist, please explain the reason;
(3) Try to explore whether there is a point on the straight line that minimizes the circumference of the straight line. If yes, find out the coordinates of the point; If it does not exist, please explain why.
(Analysis of 26 Questions in 08 Liaoning 12 City)
Solution: (1) The straight line intersects the axis at one point and intersects the axis at one point.
, 1 min
These points are all on a parabola,
The analytical formula of parabola is 3 points.
Vertex 4 o'clock
(2) There are 5 points.
7 points
9 points
(3) The score is 10.
Reason:
Solution 1:
Extend to point, make, and connect the intersection line to point, which is the required point.
1 1 min
Do something excessive.
The point is on a parabola,
In,,
, ,
In,,
, 12 point
Let the analytical formula of a straight line be
solve
13 point
solve
There are points on the straight line, which makes the circumference minimum. At this time. 14 o'clock.
Solution 2:
If the vertical line passing through the point intersects the point, the point is the symmetrical point of the point about the straight line. If the line intersects a point, the point is the desired point. 1 1.
If the intersection is the axis of the point; otherwise.
,
The same method can also be obtained.
In,,, you can get,
The median vertical line is a line segment, which can be proved to be an equilateral triangle.
Vertical division.
In other words, a point is a symmetrical point about a point. 12 points.
Let the analytical formula of the straight line be, which is obtained from the meaning of the question.
solve
13 point
solve
There are points on the straight line, which makes the circumference minimum. At this time. 14 o'clock.
85.(08 Chifeng, Inner Mongolia, 25 questions) (Full score of this question 14)
Give the following five points in the plane rectangular coordinate system.
(1) Please select three points from five points to find an analytical formula of a parabola with a straight line parallel to the axis as the symmetry axis;
(2) Find the vertex coordinates and symmetry axis of the parabola and draw a sketch;
(3) It is known that this point is on the axis of symmetry of parabola, and a straight line passes through this point and is perpendicular to the axis of symmetry. Verification: The circle centered on the center and radius is tangent to the straight line. Please further verify that the circle with the center and radius of the point on the parabola is also tangent to the straight line. What conclusion can you guess from this?
Analysis of 25 questions in Chifeng, Inner Mongolia, 25. Solution: (1) Let the analytical formula of parabola be,
Too many,
By in H.
Then. (2 points)
Get the equation,
Solve.
The analytical formula of parabola is (4 points)
(2) Pass (6 points)
Vertex coordinates are and axis of symmetry is. (8 points)
(3) (1) connection, the intersection point is perpendicular to the straight line, and the vertical foot is,
Then.
In,,,
,
,
Centered on a point, the radius is tangent to a straight line. (10)
② The vertical line connecting the intersection with a straight line has vertical feet, and the intersection has vertical feet.
Then.
In ..
.
Tangent to a straight line with a point as the center and a point as the radius. (12)
③ A circle whose center is any point on a parabola and whose radius is tangent to a straight line. (14)
86. (Question 28 in Xining, Qinghai in 2008) As shown in Figure 14, it is known that the radius of 1 intersects the axis at two points, the tangent point is, the center coordinate is, and the image of quadratic function passes through two points.
(1) Find the analytic formula of quadratic function;
(2) Find the resolution function of the tangent;
(3) Whether there is a point on the line segment that makes it a triangle similar to the vertex. If yes, request the coordinates of all qualified points; If it does not exist, please explain why.
Solution: (1) The center coordinate is, and the radius is 1, ... 1.
Image passing point of quadratic function,
Equation can get 2 points.
Solution: The second analytic function is 3 points.
(2) When the intersection is the axis, the vertical foot is .4 points.
Yes, the tangent is the tangent point (the tangent of the circle is perpendicular to the radius passing through the tangent point).
Yes,
For acute angles, 5 points will be deducted.
,
Yes,.
.
The point coordinate is 6 o'clock.
Let the tangent resolution function be, as can be seen from the meaning of the question, 7 points
The resolution function of the tangent is 8 points.
(3) existence. 9 points
(1) Take the crossing point as the axis and cross the point. You can get (two angles are equal and two triangles are similar)
, 10 point
(2) a little work, vertical foot for, a little work, vertical foot for.
Available (two angles are equal and two triangles are similar)
In,,,
In,,
, 1 1 min
The coordinates of qualified points are, 12 points.
87.(08 Qinghai Province Volume 28) Wang Liang is good at improving his learning methods. He found it best to review and reflect on the process of solving problems. One day, he spent 30 minutes studying independently. Suppose the relationship between the time (unit: minutes) he spent solving problems and his study income is shown in Figure A, and the relationship between the time (unit: minutes) he spent reviewing and reflecting on his study income is shown in Figure B (here is a parabola).
(1) Find the functional relationship between Wang Liang's learning income and problem-solving time, and write the range of independent variables;
(2) Find out the functional relationship between the learning gains of Wang Liang's review and the review time;
(3) How does Liang allocate the time for solving problems and reviewing and reflecting, so as to maximize the total learning benefit of these 30 minutes?
(The total amount of learning, the amount of learning to solve problems, and the amount of learning to review and reflect)
(08 Analysis of Qinghai Province Volume 28) Solution: (1) Hypothesis,
Substitute and get.
. ( 1)
The range of independent variables is:. (2 points)
(2) When,
Settings, (3 points)
Substitute, get,.
. (5 points)
When,
(6 points)
Namely.
(3) Let Wang Liang spend a few minutes reviewing and reflecting, and the total learning benefit is,
Then his time to solve the problem is minutes.
When,
. (7 points)
When. (8 points)
When,
. (9 points)
It decreases with the increase of,
When.
To sum up, when, at this time. (10)
In other words, when Wang Lianghua takes 26 minutes to solve the problem and 4 minutes to review and reflect, the total learning benefit is the greatest.
( 1 1)
88.(08 Shandong Jining 26 questions) (12 points)
The line segment with the length of 1cm moves to the point at the speed of 1cm/s along the edge (the point coincides with the point before moving). The line segments move at two points when they intersect the vertical line respectively, and the time for the line segments to move is s 。
(1) If the area of is, write the functional relationship of sum (write the range of independent variables);
(2) Is it possible for a quadrilateral to become a rectangle when a line segment moves? If possible, find the value at this time; If not, explain the reasons;
(3) Why are the vertices of triangles similar?
Solution: (1) When the point is at the top,
.2 points
When the point is on,
.4 points
(2) , .。
.6 points
From the condition, if the quadrilateral is rectangular, it is necessary, that is,
.
When s, the quadrilateral is a rectangle. Eight points.
(3) According to (2), when s, the quadrilateral is a rectangle. At this time,
.9 points
Besides, when, this time.
, .. 10 point
, .
Again,. 1 1.
, .
When s or s, a triangle with vertices is similar to. 12 point.
89.(08 Sichuan Bazhong 30 questions) (12 points) 30. As shown in figure 14, parabola and axis intersect at one point, one point, straight line intersects at one point, and straight line intersects with axis at one point.
(1) Write the analytical formula of the straight line.
(2) the area to be searched.
(3) If a point moves from the direction at a speed of 1 unit length per second on a line segment (it does not coincide with it), and at the same time, the point moves from the direction at a speed of 2 unit lengths per second on a ray. Let the movement time be seconds, please write the functional relationship between the areas of and, and find out how long the point has been moving and what is the maximum area.
(08 Analysis of 30 Questions in Bazhong, Sichuan) Solution: (1) In the middle, make
,
, 1 min
Click it again.
The analytical formula of is 2 points.
② Pass and get 4 points.
,
, 5 points
6 points
(3) the point is done.
7 points
8 points
Can be obtained from a straight line:
At,,, and then
, 9 points
10 point
1 1 min
This parabola opens downward when,
When the point moves for 2 seconds, its area reaches the maximum, which is. 12 point.
90. (Question 26, Zigong, Sichuan, 08) The vertex of the parabola is M, and the intersection point with the axis is A and B (point B is on the right side of point A), and the three internal angles ∠M, ∠A and ∠B of △ABM are opposite. If a quadratic equation with one variable has two equal real roots.
(1) Judge the form of △ABM and explain the reason.
(2) When the coordinate of vertex M is (-2,-1), find the analytical formula of parabola and draw the approximate figure of parabola.
(3) If the straight line parallel to the axis intersects the parabola at points C and D, and the circle with the diameter of CD is just tangent to the axis, find the center coordinates of the circle.
(08 Analysis of Question 26 in Zigong, Sichuan) Solution: (1) Order
get
Based on the inverse theorem of Pythagorean theorem and the symmetry of parabola
△ABM is an isosceles right triangle with, as the right.
(2) Settings
∫△ABM is an isosceles right triangle
The median line on the hypotenuse is equal to half of the hypotenuse.
Vertex m (-2,-1)
∴, that is, AB = 2
∴A(-3,0),B(- 1,0)
Substitute b (- 1, 0).
∴ The analytical formula of parabola is
Tuloue
(3) Let a straight line parallel to the axis be
Error in solving equation! You cannot create an object by editing the domain code.
Get, (
∴ The length of line CD is
A circle with a diameter of CD is tangent to the axis.
According to the meaning of the question
∴
solve
∴ The center coordinate is the sum.
91.(24 questions in Xinjiang autonomous region in 2008) (10) A factory will rush to produce a number of large-scale prefabricated houses for earthquake relief. As shown in the figure, the shape of one side of the prefabricated house consists of a rectangle and a parabola, the length of the rectangle is 12m, and the height of the parabolic arch is 5.6m 。
(1) Find the expression of parabola in the plane rectangular coordinate system as shown in the figure.
(2) It is necessary to install several windows in the parabolic AOB area. The bottom edge of the window is on AB. Each window is 1.5m wide and 1.6m high. The distance between adjacent windows is 0.8m, and the horizontal distance from the corner of the left and right windows to the parabola is at least 0.8m. Please calculate how many such windows can be installed at most?
Analysis of 24 Questions in Xinjiang Autonomous Region in 2008. (10) Solution: (1) Let the expression of parabola be 1.
The point is on a parabolic image.
∴
3 points
The expression of parabola is 4 points.
(2) Let the straight line above the window intersect the parabola at two points C and D, and the coordinate of point D is (k, t).
The known window height is 1.6m, ∴ 5 points.
(Give up) 6 points
∴ (Male) 7 points
There are at most n windows to install.
9 points
.
A: Up to 4 windows can be installed. 10.
This question does not require students to draw four small rectangles representing windows.