1. (Yibin City, Sichuan Province, 2008)
As shown in the figure, it is known that the parabola y=-x2+bx+c intersects the X axis and the Y axis at points A (- 1, 0) and B (0 0,3) respectively, and its vertex is d. 。
(1) Find the analytical formula of parabola;
(2) If the other intersection of the parabola and the X axis is E, find the area of the quadrilateral ABDE;
(3) Are △ AOB and △BDE similar? If similar, please prove it; If not, please explain why.
(Note: the vertex coordinates of parabola y=ax2+bx+c(a≠0) are)
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2.(08 Quzhou, Zhejiang) The position of the known right-angled trapezoidal paper OABC in the plane rectangular coordinate system is shown in the figure. The coordinates of the four vertices are O (0 0,0), A (10/0,0), B (8 8,0), C (0 0,0), and the point T is on the line segment OA (not coincident with the line segment endpoint). Fold this paper for emphasis.
(1) Find the number of times ∠OAB, and find the functional relationship between S and T when point A' is on line AB;
(2) When the figure of the overlapping part of the paper is quadrilateral, find the value range of t;
(3) Is there a maximum value for S? If it exists, find this maximum value and find the value of t at this time; If it does not exist, please explain why.
3.(08 Wenzhou, Zhejiang) As shown in the figure, the middle,,, and are the midpoint of the edge, and the point starts from the point and moves in the direction, making an intersection and making an intersection.
When a point coincides with a point, the point stops moving.
(1) Find the length of the distance from the point to the point;
(2) Find the functional relationship about (the range of independent variables is not required);
(3) Is there a point that makes it an isosceles triangle? If it exists, request all the values that meet the requirements; If it does not exist, please explain why.
4.(08 Rizhao City, Shandong Province) in △ABC, ∠ A = 90, AB = 4, AC = 3, where M is the moving point on AB (not coincident with A and B), passing through M is MN‖BC, AC is at N, with MN as the diameter ⊙O and at ⊙.
The area s of (1)△MNP is expressed by an algebraic expression with x;
(2) When what is the value of x, ⊙O is tangent to the straight line BC?
(3) In the process of moving point M, remember that the overlapping area of △MNP and trapezoidal BCNM is y, try to find the functional expression of y about x, and find the value of x and the maximum value of y?
5. (Zhejiang Jinhua, 2007) As shown in figure 1, hyperbola y =(k >；; 0) The straight line y = k ′ x intersects at point A and point B, and point A is in the first quadrant. Try to solve the following problems: (1) If the coordinate of point A is (4,2), the coordinate of point B is; If the abscissa of point A is m, the coordinates of point B can be expressed as:
(2) As shown in Figure 2, make a straight line L after crossing the origin O, and cross the hyperbola y =(k>；; 0) at p and q, point p is in the first quadrant. ① indicates that the quadrilateral APBQ must be a parallelogram; ② the abscissas of points a and p are m and n, respectively. Can quadrilateral APBQ be a rectangle? Will it be a square? If possible, directly write out the conditions that mn should meet; If not, please explain why.
6. (Jinhua, Zhejiang, 2008) As shown in Figure 1, in the plane rectangular coordinate system, it is known that AOB is an equilateral triangle, the coordinates of point A are (0,4), point B is in the first quadrant, and point P is the moving point on the X axis. Connect AP, and rotate AOP counterclockwise around point A to make the edges AO and AB coincide, thus obtaining Abd. (2) When point P moves to point (0), find the length of DP at this time and the coordinates of point D; (3) Whether there is a point P, so that the areas of Δ δOPD are equal, and if there is, the coordinates of the point P meeting the requirements are requested; If it does not exist, please explain why.
7. (Yiwu, Zhejiang, 2008) As shown in figure 1, the quadrilateral ABCD is a square, and G is the moving point on the edge of CD (the G point does not coincide with C and D). Take CG as one side and make a square CEFG outside the square ABCD to connect BG and de. We explore the length relationship between line segment BG and line segment DE and the position relationship of straight line in the following figure:
(1)① guess the length relationship between line segment BG and line segment DE and the position relationship of straight line as shown in figure1;
② Rotate the square CEFG in figure 1 clockwise (or counterclockwise) at any angle around point C, and get the situation as shown in figures 2 and 3. Please observe and measure whether the conclusion drawn from diagram 1 is still valid, and choose Figure 2 to prove your judgment.
(2) The square in the original problem is changed into a rectangle (as shown in Figure 4-6), AB=a, BC=b, CE=ka, CG=kb (a b, k 0). Which conclusions are valid and which are not? If so, take Figure 5 as an example to briefly explain the reasons.
(3) In Figure 5 of the problem (2), connect,, and a=3, b=2, k=, and evaluate.
8. (Yiwu, Zhejiang, 2008) As shown in figure 1, the vertices A and C of the right-angled trapezoidal OABC are on the positive and negative semi-axes of the Y axis respectively. After passing through point B and point C, the straight line translates, and the translated straight line intersects the axis of point D and the axis of point E. 。
(1) translate the straight line to the right, let the translation distance CD be (t 0), and the area swept by the straight line (the shaded part in the figure) be. The correlation function image is shown in Figure 2. OM is a line segment, MN is a part of a parabola, NQ is a ray, and the abscissa of n points is 4.
① Find the length of the trapezoid upper bottom AB and the area of the right-angled trapezoid OABC;
(2) When, find the resolution function of S;
(2) Under the condition of the problem (1), when the straight line moves to the left or right (including overlapping with the straight line BC), is there a point P on the straight line AB, which makes it an isosceles right triangle? If it exists, please directly write the coordinates of all points p that meet the conditions; If it does not exist, please explain why.
9. (Yantai, Shandong Province, 2008) As shown in the figure, the side length of rhombic ABCD is 2, BD=2, E and F are two moving points on the sides of AD and CD respectively, and AE+CF=2.
(1) Verification: △ BDE △ BCF;
(2) Judge the shape of △BEF and explain the reason;
(3) Let the area of △BEF be S and find the range of S. 。