Elliptic knowledge point summary
1, the concept of ellipse
The locus (or set) of a point whose sum of the distances from a point to two fixed points F 1 and F 2 on a plane is equal to a constant (greater than | F 1 F 2 |) is called an ellipse, and these two fixed points are called the focus of the ellipse, and the distance between the two focuses is called the focal length.
Let p = {m || mf 1 | | mf2 | = 2a}, | f 1f2 | = 2c, where a > 0, c > 0, and a and c are constants:
(1) if a > c, then the set p is an ellipse;
(2) If a = c, the set p is a line segment;
(3) If a < c, the set p is empty.
2. Standard equation and geometric properties of ellipse.
A rule
The relationship between the position of ellipse focus and X 2, Y 2 coefficients;
Two methods
(1) Definition method: According to the definition of ellipse, determine the values of a 2 and B 2, and then write the ellipse equation directly in combination with the focus position.
(2) undetermined coefficient method: according to whether the focus of the ellipse is on the X axis or the Y axis, set the corresponding standard equation, and then determine the' equation about A, B and C according to the conditions, and solve a 2 and b 2, thus writing the standard equation of the ellipse.
Three skills
(1) Of all the distances from any point M on the ellipse to the focus F, the distances from the endpoint of the long axis to the focus are the maximum distance and the minimum distance, respectively, with the maximum distance being A+C and the minimum distance being A-C. ..
(2) To find the ellipse eccentricity e, we only need to find the homogeneous equations of a, b and c, and then combine b 2 = a 2-c 2 to get e (0< e < 1+0).
(3) When solving elliptic equations, the undetermined coefficient method is often used, but first, it is necessary to judge whether it is a standard equation. The basis of judgment is:
(1) whether the center is at the origin;
② Whether the axis of symmetry is the coordinate axis.
Second, review the guidance.
1, master the definition of ellipse and its geometric properties, and you will find the standard equation of ellipse.
2. Master several common mathematical thinking methods-function and equation, combination of numbers and shapes, transformation and reduction, and understand the essential problems of analytic geometry-solve geometric problems by algebraic methods.