I compiled a book "Mathematics for Freshmen in College Entrance Examination", which is about a conic curve with four lines and one equation.
1, if P(x0, y0) is on the ellipse x2/a2+y2/B2 = 1, the tangent equation is as follows.
x0x/a2+y0y/B2 = 1；
If P(x0, y0) is outside the ellipse x2/a2+y2/B2 = 1, the tangent chord equation is as follows
x0x/a2+y0y/B2 = 1；
These two equations have the same form but different meanings. PPPPPP2
2. If P(x0, y0) is on the hyperbola x2/a2-y2/b2 = 1, the tangent equation is as follows.
x0x/a2-y0y/B2 = 1；
If P(x0, y0) is outside the ellipse x2/a2-y2/b2 = 1, the tangent chord equation is as follows
x0x/a2-y0y/B2 = 1；
3. If P(x0, y0) is on the parabola y2=2px, the tangent equation is as follows
y0y = p(x0+x)；
If P(x0, y0) is outside the parabola y2=2px, the tangent chord equation is as follows
y0y = p(x0+x)；
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Question 2: What are the skills for solving conic curves? Two definitions of 1. conic curve;
In the first definition of (1), we should pay attention to the restrictions in parentheses: in an ellipse, the sum of the distances to two fixed points F and F is equal to a constant, which must be greater than, when the constant is equal to, the trajectory is a line segment F F, and when the constant is less than, there is no trajectory; In hyperbola, the absolute value of the difference between the distances of two fixed points f and f is equal to a constant, and this constant must be less than |F F |. "Absolute value" and < | f f | in the definition cannot be ignored. If = |F F |, the trajectory is two rays with f and f as endpoints; If | f f |, the trajectory does not exist. If the absolute value in the definition is removed, the trajectory only represents a hyperbola.
If the curve represented by the equation is _ _ _ _ (A: the left branch of hyperbola)
(2) In the second definition, it should be noted that the fixed point and the fixed line are the corresponding focus and directrix, and the quotient of "the distance between points is the numerator and the distance between points and lines is the denominator" is the eccentricity. The second definition of conic gives the relationship between the distance from the point to the focus and the distance from this point to the corresponding directrix, so we should be good at transforming them into each other by using the second definition.
If the point on the parabola and a moving point P(x, y) are known, the minimum value of y+|PQ| is _ _ _ (answer 2).
2. Standard equation of conic curve (standard equation refers to the equation of standard position when the center (vertex) is at the origin and the coordinate axis is the axis of symmetry);
(1) ellipse: when the focus is on the axis (), when the focus is on the axis = 1 (). What is the necessary and sufficient condition for an equation to represent an ellipse? (ABC≠0, and the numbers of A, B and C are the same, A≠B).
If (1) is known that the equation represents an ellipse, the value range of is _ _ _ _ (a:);
(2) If sum, the maximum value of is _ _, and the minimum value is _ _ (A:).
(2) Hyperbola: focus axis: = 1, focus axis: = 1 (). What is the necessary and sufficient condition for an equation to represent a hyperbola? (ABC≠0, the numbers of A and B are different).
If the center of the circle is at the coordinate origin, the focus is on the coordinate axis, and the hyperbola C of eccentricity passes through this point, then the equation of C is _ _ _ _ _ _ _ (a:).
(3) Parabola: when the opening is to the right, when the opening is to the left, when the opening is up and when the opening is down.
If the two endpoints of a line segment AB with a fixed length of 3 move on y=x2, and the midpoint of AB is m, find the shortest distance from the point m to the X axis.
3. Judgment of the focus position of the conic (first converted into the standard equation, then judged):
(1) ellipse: It is determined by the size of the denominator, and the focus is on the axis with the largest denominator.
If the equation is known to represent an ellipse whose focus is on the Y axis, the range of m is _ _ (A:).
(2) Hyperbola: it is determined by the positive and negative coefficients of the term, and the focus is on the coordinate axis of the positive coefficient;
(3) Parabola: The focus is on the coordinate axis of the primary term, and the sign of the primary term determines the opening direction.
Special reminder: (1) When solving the problems of ellipse and hyperbola, we must first judge the position of the focus. The positions of focus F and F are the positioning conditions of ellipses and hyperbolas, which determine the types of standard equations of ellipses and hyperbolas. Two parameters in the equations determine the shapes and sizes of ellipses and hyperbolas, which are the forming conditions of ellipses and hyperbolas. When solving parabolic problems, we must first judge the opening direction; (2) In an ellipse, it is the largest, and in a hyperbola, it is the largest.
4. Geometric properties of conic section:
(1) ellipse (take () as an example): ① Range:; 2 key points: two key points; ③ Symmetry: two symmetrical axes, one symmetrical center (0,0) and four vertices, in which the length of major axis is 2 and the length of minor axis is 2; (4) Alignment: two alignment lines; ⑤ Eccentricity: ellipse, the smaller the ellipse, the more round the ellipse; The bigger the ellipse, the flatter it is.
For example, (1) If the eccentricity of an ellipse is _ _ (A: 3 or);
(2) When the maximum area of a triangle with one point on the ellipse and two focal points of the ellipse as vertices is 1, the minimum value of the long axis of the ellipse is _ _ (A:).
(2) Hyperbola (take () as an example): ① Range: or; 2 key points: two key points; ③ Symmetry: two symmetrical axes, a symmetrical center (0,0) and two vertices, where the real axis length is 2 and the imaginary axis length is 2. Especially when the lengths of the real axis and the imaginary axis are equal, it is called equilateral hyperbola, and its equation can be set as: (4) Linearity: two linear lines; ⑤ Eccentricity: hyperbola, equilateral hyperbola, yue ... >>
Question 3: What are the six famous circles of conic section and their properties? Unified definition of conic curve: (second definition)
The distance between a plane and a fixed point (focus) and a fixed line (directrix) is the * * * of a point with a constant eccentricity (e). According to the size of e, it can be divided into ellipse, parabola and hyperbola. A circle can be regarded as a curve, and e is 0.
1.0x^2/a^2+y^2/b^2= 1(0y^2/a^2+y^2/b^2= 1(0a^2=b^2+c^2
The sum of the distances from any point on an ellipse to two focal points is 2a (fixed value), which is greater than the focal length 2c. This is the first definition.
Question 4: Who can tell me what games are in beta now? Go to 17 173.