When it comes to the relationship between conic curve and chord, this formula often works well. In addition, this method is suitable for all kinds of conic curves, including circles, ellipses, parabolas and hyperbolas.
The slip method and some common conclusions derived from the slip method belong to high-frequency test sites in college entrance examination mathematics, so we must pay attention to them.
Represents two different points on an ellipse.
Subtract the two expressions to get:
Of course, it can also be written as:
Where represents the midpoint of the chord.
Explanatory formula
The above formula can be explained in words as follows:
This is an important and commonly used conclusion, and it is also a high-frequency test center.
Example of real question
20 15 national volume 2 directly takes the derivation process of the above common conclusions as the test questions. See:
20 15 National Volume B Question 20
There are still more problems that need to be solved by applying the above conclusions:
20 10 national examination paper topic 20
20 10 national papers on mathematics 20
20 13 national volume question b question 20
2020 National Science Mathematics Volume A Question 20
Parabolic equation:
Because the two points are on a parabola,
,
If the midpoint is, then
Or:
Explanatory formula
The above formula can be expressed in words as follows:
For parabola with axis as symmetry axis, the following conclusion holds:
(1) The product of the slope of the chord of a parabola and the coordinates of the midpoint of the chord is equal to the focal length.
(2) The same set of parallel chords (with equal slopes), in which the points are on the same straight line perpendicular to the axis.
(3) According to the slope of the chord of the parabola, the coordinates of the chord can be calculated; or vice versa, Dallas to the auditorium
Example of real question
20 18 national mathematics volume b question 20
20 17 national science mathematics volume c topic 20
1987 national examination question 2 1
Parabolic equation:
Because the two points are on a parabola,
,
If the midpoint is, then
Or:
Explanatory formula
The above formula can be expressed in words as follows:
Explanatory formula
The above formula can be expressed in words as follows:
For parabola with axis as symmetry axis, the following conclusion holds:
(1) The product of the slope of the chord of a parabola and the coordinates of the midpoint of the chord is equal to the focal length.
(2) The same set of parallel chords (with equal slopes), in which the points are on the same straight line perpendicular to the axis.
(3) According to the slope of the chord of the parabola, the coordinates of the chord can be calculated; or vice versa, Dallas to the auditorium
Example of real question
20 17 national volume a question 20
If the equation of a circle is:
There are two points on the circle, and the midpoint is, then
In other words, in fact, the vertical diameter theorem is obtained by analytical method.
As shown in the figure, the parabolic equation is:, which is the chord of the parabola. If the slope of the chord remains the same, it moves to the left, in which the coordinates of the points remain the same, and at the same time, the three points are getting closer and closer, and finally become a point. At this time, the straight line and parabola have only one thing in common, and the straight line will also change from the chord of parabola to the tangent.
In other words, if you make a chord parallel to the tangent, the coordinates of the midpoint of the chord are equal to the coordinates of the tangent point.
If the tangent coordinate is, then
The equation of tangent is:
Similarly, if the equation of parabola is:, then
The equation of tangent is:
What can the square difference method do?
The function of the square difference method (point difference method) is simply to link the slope of the chord with the midpoint coordinates of the chord, and there are many problems that can be solved:
(1) chord length problem
(2) Find the locus equation of the midpoint of the chord.
(3) Find the slope range of the chord.
(4) Find the tangent equation.
(5) Fixed-point problem
As can be seen from the previous real questions, there are many opportunities for this method to be used in the college entrance examination.