(1) Find the coordinates of this point;
(2) Find the function expression of parabola;
(3) links. Is there a point on the axis that makes the triangle with this point as its vertex similar? If yes, request the coordinates of the point; If it does not exist, please explain why.
[Solution] A straight line intersects an axis at a point, when,
The coordinates of this point are. A parabola passes through two points on the axis.
And the symmetry axis is, according to the symmetry of parabola, the coordinates of the point are.
(2) exaggerated, easy to understand.
Parabolic again,
Solve.
(3) Link, from, from,
Let the symmetry axis of parabola intersect at one point,
It's easy to get from the main points,
In isosceles right triangle, it is obtained by Pythagorean theorem.
Suppose there is a point on the axis, so a triangle with a point as its vertex is similar to.
When, when,.
That is, the point coincides with the point, and the coordinates are.
(2) When and when.
That is, ...,
The coordinates of are.
.
The point cannot be located on the axis to the right of the point.
To sum up, there are two points on the axis, which can make the triangle with this point as the vertex similar.
2. The image of quadratic function (Henan Volume) is shown in the figure. The straight line passing through a point on the axis intersects with the parabola, and these two points and the passing point are respectively taken as the vertical lines of the axis, and the vertical feet are respectively.
(1) When the abscissa of a point is, find the coordinates of the point;
(2) In the case of (1), judge whether there is a point on the axis, the axis and the axis respectively, and make them at right angles. If yes, find the coordinates of the point; If it does not exist, please explain the reason;
(3) When a point moves on a parabola (the point does not coincide with the point), it is evaluated.
[Solution] (1) According to the meaning of the question, the coordinate of the point is, where the abscissa of the point is,, axis, axis,, ... that is.
Solve (give up).
(2) existence.
Links,.
By (1),,, and then set.
Axis, axis,,.
Are the solutions of the original equation.
The coordinates of this point are or.
(3) according to the meaning of the question, set,, might as well set,.
According to (1),
And then still.
Simplify and acquire.
,
.
.
3. (Zhanjiang, Hubei) It is known that the parabola intersects the axis at a point, which is the two real roots of the equation, and the point is the intersection of the parabola and the axis.
(1).
(2) The analytical expressions of the sum of straight lines are obtained respectively;
(3) If the moving straight line and the line segment intersect at two points respectively, is there a point on the axis that makes it an isosceles right triangle? If it exists, find out the coordinates of the point;
[Solution] (1) from and from.
Substitute the coordinates of the two points into the simultaneous solution respectively, and get
.
(2) Available from (1), when, …
Set, respectively, into two coordinates, at the same time.
The analytical formula of the straight line is.
In the same way, the analytical formula of straight line can be obtained as follows.
(3) Assuming that there are points that meet the conditions, let the intersection of the straight line and the axis be.
(1) When they are waist, take the passing point as the axis, as shown in the figure, and the sum is an isosceles right triangle.
,
.
, ,
Which is the solution.
The ordinate of the point is that the point is on a straight line,
, for answers,.
In the same way.
(2) When it is the bottom edge,
The midpoint of the intersection is the axis of the point, as shown in the figure.
Then,
By,
Yes, that is, yes.
Using the method of 1,
, .
According to the chart,
Yes, it also meets the requirements.
To sum up, there are three * * * eligible, namely
4. In the rectangle, establish a rectangular coordinate system with the coordinate origin and the straight line as the axis. Then rotate the rectangle counterclockwise around the point so that the point falls on the axis, and the summation point falls on the point and axis in the second quadrant in turn (as shown in the figure).
(1) Find the quadratic resolution function passing through three points;
(2) Let the straight line intersect with the quadratic function image of (1) at another point to find the perimeter of the quadrilateral.
(3) Let a point on the quadratic function image be (1) and find the coordinates of this point.
(1) solution: According to the meaning of the question,,.
, , .
Suppose the second resolution function after three points is.
Substitution, get 0.3 points.
The second analytic function sought is:
.
(2) Solution: According to the meaning of the question, the quadrilateral is a rectangle.
And ...
The coordinates of the intersection of the straight line and the quadratic function image are,
.
With respect to parabolic symmetry,
.
Perimeter of quadrilateral
.
(3) Set the intersection axis to.
, namely
, so.
Let the analytical formula of a straight line be.
Substitute, substitute,
Get a solution.
constitutive equations
Solution or (this set of numbers is point coordinates)
The required point coordinates are.