Laws and methods:

The definition of 1. conic curve is the source of the corresponding standard equation and geometric properties. For the problems related to conic curve, we should have the consciousness of using conic curve definition to solve problems, and "regression definition" is an important problem-solving strategy.

2. When studying the maximum distance between points, we usually define the distance from a point on a curve to a focus to another focus, or convert the distance from a point to a focus on a curve to the corresponding quasi-line by definition, and then solve the maximum problem with the idea of combining numbers with shapes.

Example 1 If point M(2, 1) and point c are ellipses X2 16+Y2.

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Right focus = 1, and point A is the moving point of an ellipse, then the maximum value of | AM |+| AC |.

The small value is _ _ _ _ _

Tracking training 1 known ellipse X29+Y2

5 = 1, F 1 and F2 are the left and right focal points of the ellipse respectively, and point A (1, 1) is a point inside the ellipse.

Point p is a point on an ellipse, and find the maximum value of | pa |+| pf 1 |.

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Question 2: Questions about the properties of conic curves.

Law and method

The focus, eccentricity and asymptote of conic are common problems in the exam. As long as you master the basic formulas and concepts and fully understand the meaning of the questions, most of them can be solved smoothly.

Example 2: Ellipse X23m2+Y25N2 = 1 and hyperbola X22m2-Y2 are known.

3N2 = 1 has a common focus, then the asymptote of hyperbola.

The equation is

Tracking training 2 knows that the eccentricity of hyperbola X2A2-Y2B2 = 1 is 2, and the focus is with ellipse X225+Y2.

The focus of 9 = 1 is the same, then

The focal coordinate of unary hyperbola is _ _ _ _ _ _ _; Asymptote equation is _ _ _ _ _.

The relationship between the position of three straight lines and conic curve

Laws and methods:

1. The positional relationship between straight lines and conic curves can be divided into three categories: no common point, only one common point and two different common points. There is only one common point between the straight line and the conic curve, that is to say, the straight line is tangent to it as an ellipse. For hyperbola, it means that it is tangent to it or the straight line is parallel to the asymptote of hyperbola; For parabola, it refers to a straight line tangent to it or parallel to its axis of symmetry.

2. Questions about the positional relationship between straight line and conic curve may involve chord length, focus chord, midpoint of chord, range of values, maximum value and so on.

This kind of problem is comprehensive. To analyze this kind of problems, we often use the idea of combining numbers with shapes, the method of "setting without seeking", the symmetry method and the relationship between roots and coefficients.

Example 3 Known ellipse C: X2A2+Y2B2 =1(a >; B>0) has an eccentricity of 6.

3. The distance from one end of the short axis to the right focus is 3.

(1) Find the equation of ellipse c;

(2) Let the straight line L and ellipse C intersect at point A and point B, and the distance from the coordinate origin O to the straight line L is 3.

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, find the maximum value of △AOB area.

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In tracking training 3, vectors A = (x, 3y), B = (1, 0) and (a+3b) ⊥ (a-3b). (1) Find the equation of locus c of point Q(x, y);

(2) Let curve c and straight line Y = KX+M intersect at two different points m and n, and point A(0,-1). When | AM | = | An |, the value range of the number m is realistic.

Question 4: Trajectory problems related to conic curves.

Laws and methods:

Trajectory is formed by the movement of moving points according to certain laws, and the conditions of trajectory can be expressed by the coordinates of moving points. The basic method to solve the trajectory equation is as follows

(1) directly solve the trajectory equation: establish a suitable rectangular coordinate system and list the equations according to the conditions; (2) Solving the trajectory equation by the undetermined coefficient method: according to the standard equation of the curve; (3) Solving the trajectory equation by definition method: the trajectory of the moving point satisfies the definition of conic curve;

(4) Solving the trajectory equation by substitution method: the moving point M(x, y) depends on the coordinate changes of the points (x0, y0) on the known curve C. According to the relationship between them, the relationship of x, y, x0, y0 is obtained, and x0, y0 are represented by x, y, and substituted into the equation of curve C. Example 4 shows that the known line segment AB = 4, moving circle.