The locus of a moving point whose sum of distances to two fixed points F 1 and F2 is constant (greater than |F 1F2 |) is called an ellipse.
2. The standard equation of ellipse is:
3. When the focus is on the X axis, the relationship between A, B and C in the ellipse is a2=b2+c2.
When the focus is on the y axis
1, range:
-a ≤ x ≤ a，-b ≤ y ≤ B。
The ellipse falls in a rectangle consisting of x = a and y = b.
Symmetry of ellipse
2. Symmetry:
Graphically, the ellipse is symmetrical about the X axis, the Y axis and the origin.
Look from the equation:
(1) Replace x with -x equation, and the image is symmetrical about y;
(2) change y into -y equation, and the image is symmetrical about x;
(3) Change X to -x, and change Y to -y equation at the same time. The image is symmetrical about the origin center.
3, the vertex of the ellipse
Let x=0 and get y=? This explains the intersection of the ellipse and the y axis?
Let y=0 and get x=? Explain the intersection of ellipse and x axis?
* Vertex: The four intersections of an ellipse and its axis of symmetry are called the vertices of an ellipse.
* Major axis and minor axis: Line segments A 1A2 and B 1B2 are called major axis and minor axis of ellipse respectively.
A and b are called the major axis and minor axis of an ellipse respectively.
Draw the following figure (1) (2) a1a2b2b1a1according to the previous knowledge.
4. Eccentricity of ellipse
Eccentricity: the ratio of the focal length of an ellipse to the length of its major axis;
This is called eccentricity of ellipse.
[1] Range of eccentricity:
[2] Influence of eccentricity on ellipse shape: 0
2) The closer E is to 0, the closer C is to 0, so the bigger B is, the more round the ellipse is.
[3] the relationship between e and a and b:
|x|≤ a，|y|≤ b
Axisymmetric about X axis and Y axis; Symmetry about the origin center
(a，0)，(a，0)，(0，b)，(0，-b)
(c，0)、(-c，0)
The long semi-axis length is a, and the short semi-axis length is b.a >; ba2=b2+c2|x|≤ a，|y|≤ b
Axisymmetric about X axis and Y axis; Symmetry about the origin center
(a，0)，(a，0)，(0，b)，(0，-b)
(c，0)、(-c，0)
The long semi-axis length is a, and the short semi-axis length is b.a >; Ba2=b2+c2|x|≤ b, |y|≤ a Same as above (b, 0), (-b, 0), (0, a), (0, -a).
(0, c), (0, -c) Same as before, same as before, same as before, 1 The known elliptic equation is 16x2+25y2=400.
The length of its major axis is:. The length of the minor axis is:.
The focal length is:. Eccentricity equals.
The focal coordinates are:. Vertex coordinates are:.
The area of the circumscribed rectangle is equal to. 108680
2. Determine the position of the focus and the position of the long axis.
The elliptic equation is called 6x2+y2=6.
The length of its major axis is:. The length of the minor axis is:.
The focal length is:. Eccentricity is equal to.
The focal coordinates are:. Vertex coordinates are:.
The area of the circumscribed rectangle is equal to. 2 exercise 1.
Example 3. It is known that the center of the ellipse is at the origin, the focus is on the coordinate axis, the long axis is three times as long as the short axis, and the ellipse passes through point p (3 3,0), so the equation of the ellipse is found.
Summary of mathematical thought of classified discussion: In this lesson, we learned several simple geometric properties of ellipse: range, symmetry, vertex coordinates, eccentricity and other concepts and their geometric significance. We know several basic quantities A, B, C, E, vertex, focus, center of symmetry and their relationships, which is very helpful for us to solve related problems in ellipses and lays a solid foundation for us to learn the other two kinds of conic curves in the future. In the study of analytic geometry, we explore the implicit conditions in the topic from the perspective of equation form, which requires us to skillfully understand and master the relationship between number and shape. In this lesson, we use geometric properties and undetermined coefficient method to solve elliptic equations. In the process of solving problems, the mathematical ideas of function equation and classification discussion are accurately reflected.