The trajectory of a point on a plane, whose distance is equal to the distance between a fixed point and an alignment, is called a parabola. The fixed point is called the focus of parabola, and the fixed line is called the directrix of parabola.

Note: (1) The fixed point is not on this straight line;

(2) What is the trajectory of a fixed point on this straight line?

2. Derive the standard equation of parabola:

(1) The focus of the parabola it represents is on the positive semi-axis of the shaft, and the focal coordinates are,

Its directrix equation is

(2) There are four different cases of parabola, because its position in the coordinate system is different and its equation is different, so there are several other forms of standard equation of parabola:,,. The graphs, standard equations, focal coordinates and directrix equations of these four parabolas are as follows.

3. Parabolic directrix equation: As shown in the figure, establish rectangular coordinate systems respectively, and set (), then the standard equation of parabola is as follows:

Focus coordinates of standard equation graph

collinearity equation

Opening direction

Same point: (1) parabolas all pass through the origin;

(2) The symmetry axis is the coordinate axis;

(3) The directrix is perpendicular to the axis of symmetry, and the vertical foot and the focus are symmetrical about the origin on the axis of symmetry; Their distance from the origin is equal to the absolute value of the first term coefficient, that is;

Difference: (1) When the graph is symmetrical, it is a linear term and a quadratic term.

The right end of the equation is, and the left end is;

When the graph is symmetrical, it is a quadratic term and a linear term.

The right end of the equation is, and the left end is.

(2) When the opening direction is in the positive direction of the shaft (or shaft), the focus is on the positive semi-axis of the shaft (or shaft), and the right end of the equation takes a positive sign;

When the opening is in the negative direction of the axis (or shaft) and the focus is on the negative semi-axis of the axis (or shaft), the right end of the equation takes a negative sign.

Four. Applied mathematics:

Example 1 (1) It is known that the standard equation of parabola is the equation for finding its focal coordinates and directrix.

(2) Given that the focal coordinate of parabola is (0, -2), find its standard equation.

Analysis: There is only one parameter p in the standard equation of parabola. So as long as the standard form of parabola is determined, we can write the equation by finding the value of p, but we should pay attention to the situation of two solutions.