High-frequency test center of college entrance examination mathematics: standard equation of ellipse

The standard equation of 1. ellipse * * * is divided into two cases:

When the focus is on the X axis, the standard equation of ellipse is: x2/a2+y2/b2= 1, (a >；; b & gt0);

When the focus is on the Y axis, the standard equation of ellipse is: y2/a2+x2/b2= 1, (a >；; b & gt0);

2. Let the two foci of the ellipse be F 1 and F2 respectively, and the distance between them is 2c. The sum of the distances from any point on the ellipse to F 1 and F2 is 2a (2a >: 2c).

3. Equation geometric properties of elliptic circle

The range of x and y

When the focus is on the X axis, -A ≤ X ≤ A,-B ≤ Y ≤ B.

When the focus is on the y axis, -B ≤ X ≤ B,-a ≤ y ≤ a.

symmetrical

No matter whether the focus is on the X axis or the Y axis, the ellipse is always symmetrical about the X/Y/ origin.

Vertex:

When the focus is on the X axis: the vertex of the long axis: (-a, 0), (a, 0)

Vertex of minor axis: (0, b), (0, -b)

When the focus is on the Y axis, the vertices of the long axis are (0, -a), (0, a).

Vertex of minor axis: (b, 0), (-b, 0)

Pay attention to which axis the long and short axes represent respectively, which is easy to cause confusion here and needs to be understood step by step in combination with numbers and shapes.

Key points:

When the focus is on the X axis, the focus coordinate F 1(-c, 0)F2(c, 0).

When the focus is on the Y axis, the focus coordinate F 1(0, -c)F2(0, c).

4.S=πab (where A and B are the lengths of the major axis and minor axis of the ellipse respectively, which can be deduced from the area of the circle) or S=πAB/4 (where A and B are the lengths of the major axis and minor axis of the ellipse respectively).

5. Relationship between circle and ellipse: ellipse includes circle, and circle is a special ellipse.