When the focus is on the X axis, the standard equation of the ellipse is: X 2/A 2+Y 2/B 2 =1,(a>b>0);

When the focus is on the Y axis, the standard equation of the ellipse is: Y 2/A 2+X 2/B 2 =1,(a>b>0);

Where a 2-c 2 = b 2.

Deduction: pf1+pf2 >; F 1F2(P is the point f on the ellipse as the focus)

Chinese name

standard equation of the ellipse

Foreign name

standard equation of the ellipse

Another name

line

express

x^2/a^2+y^2/b^2= 1

presenter

mathematician

Equation derivation

Let the two foci of an 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).

Taking the straight line of F 1 and F2 as the X axis and the vertical line of the line segment F 1F2 as the Y axis, and establishing the rectangular coordinate system xOy, the coordinates of F 1 and F2 are (-c, 0) and (c, 0) respectively.

Let M(x, y) be any point on the ellipse, and we can know from the definition of ellipse.

|MF 1|+|MF2|=2a，(a & gt0)

that is

At the same time, find the square of both sides of the equation and simplify it.

Then square the two sides and simplify them.

and

, setting

Have to

Divide both sides by the same number and you will get

This form is the standard equation of ellipse.

It is generally believed that a circle is a special case of an ellipse [2].

Nonstandard equation

Its equation is a binary quadratic equation, and its characteristics can be calculated and analyzed by using the properties of the binary quadratic equation [3].

Geometric attribute

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 thoroughly by combining numbers [4].

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).

Calculation method

((where 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 (where are the major axis and minor axis of the ellipse respectively) [5].

The relationship between a circle and an ellipse:

Ellipses include circles, and circles are special ellipses.

reference data

[1] Cao. Encyclopedia of Middle School Teaching in China: Mathematics Volume [M]. Shenyang: Shenyang Publishing House

[2] Shen Jinxing. Derivation of Elliptic Standard Equation from the Perspective of Mathematical Culture [J]. Mathematical Communication, 20 15(8).