Teaching requirements: abstract the model of ellipse from specific situations and master the definition and standard equation of ellipse.

Teaching Emphasis: Definition of Ellipse and Standard Equation

Teaching Difficulties: Derivation of Elliptic Standard Equation

Teaching process:

First, the new lesson introduction:

Take a string with a fixed length, fix its two ends at the same point on the drawing board, put on a pencil, tighten the rope and move the nib, then the trajectory drawn by the nib is a circle. If the two ends of the string are separated by a certain distance, fixed at two points on the drawing board, put on a pencil, tighten the rope and move the pen tip, what curve will be drawn? (Students do it and observe the results)

Thinking: What are the geometric conditions of the moving pen tip (moving point)?

Thinking after observation: In the process of moving the pen tip, the length of the string remains unchanged, that is, the sum of the distances from the pen tip to two fixed points is equal to a constant.

Second, teach new lessons:

1. Define an ellipse: the trajectory of a point whose sum of distances to two fixed points on a plane is equal to a constant (greater than) is called an ellipse. These two fixed points are called the focal points of the ellipse, and the distance between the two focal points is called the focal length of the ellipse.

2. Derivation of elliptic standard equation;

A rectangular coordinate system is established with the straight line passing through the two focal points of the ellipse as the axis and the middle vertical line of the line segment as the axis. Let it be any point on the ellipse, and the focal length of the ellipse is, then the coordinate of the focal point is, and the sum of the distances from is equal to. According to the definition of ellipse, yes, if you substitute it with the distance formula between two points, it will be clear at this time. In other words, the standard equation of an ellipse is. According to symmetry, if the focus is on the axis, the ellipse is.

Through the definition and derivation of ellipse, we emphasize two basic equations to students: and.

3. Example 1 Write the standard equation of an ellipse that meets the following conditions: (1) The focus is on the axis;

(2), the focus is on the axis; (3) (Teachers' Guidance-Students' Answers)

Example 2 Given that the coordinates of the two focal points of an ellipse are the sum crossing point, find its standard equation.

(Teacher Analysis-Student Performance Board-Teacher Comments)

Third, consolidate the exercises:

1. Write the standard equation of ellipse applicable to the following conditions:

(1) The focal point is on the axis, and the focal length is equal to and passes through this point;

(2) The focal coordinates are:

⑶ .

2. Homework: Question 2.

Chapter 2. 1.2 ellipse and its standard equation

Teaching requirements: master the solution of point trajectory, the basic idea and application of coordinate method.

Teaching emphasis: finding the locus equation of points, the basic idea and application of coordinate method.

Teaching difficulties: finding the locus equation of points, the basic idea and application of coordinate method.

Teaching process:

First, let's review:

1. Definition of ellipse, focal coordinates and focal length of ellipse.

2. Two basic equations about ellipse.

Second, teach new lessons:

1. Example 1 The coordinates of the point are,. When a straight line intersects a point, the product of its slopes is the trajectory equation of the point.

Find the trajectory of which point, set the coordinates of which point, and then find the related equation containing points.

(Teacher's guidance-demonstration writing)

2. Exercise: 1. The coordinates of a point are the intersection of a straight line and the point, and the quotient of the slope of the straight line and the slope of the straight line is, what is the trajectory of the point?

(Teacher Analysis-Student Performance Board-Teacher Comments)

2. Use the ratio of the fixed point to the distance to the fixed line to find the trajectory equation of the fixed point.

(Teacher Analysis-Student Performance Board-Teacher Comments)

3. Example 2 takes any point on the circle, and the vertical line segment passing through this point as the axis is the vertical foot. When a point moves on a circle, what is the trajectory of the midpoint of the line segment?

Correlation point method: find the relationship between coordinates and the midpoint of a point, and then eliminate it to get the trajectory equation of the point.

(Teacher's guidance-demonstration writing)

Exercise:

1. Question 7.

2. Given that one side of a triangle has a length of and a perimeter of, find the trajectory equation of the vertex.

5. Knowledge summary:

(1) pay attention to find the trajectory of which point, set the coordinates of which point, and then find out the equation related to the point.

② Associated point method: Find the relationship between coordinates and the midpoint of a point, and then eliminate it to get the trajectory equation of the point.

Third, homework:

Question 4

Practice 8: Refinement and refinement.

Chapter 2.2 Simple Geometric Properties of Ellipse

Teaching requirements: according to the equation of ellipse, study the geometric properties of curve and draw its figure correctly; According to the geometric conditions, the curve equation is obtained, and its properties are studied and drawn by using the curve equation.

Teaching emphasis: solving elliptic equations through geometric properties and drawing.

Difficulties in teaching: drawing elliptic equations through geometric properties and diagrams.

Teaching process:

First, let's review:

1. Definition of ellipse, focal coordinates and focal length of ellipse.

2. The standard equation of ellipse.

Second, teach new lessons:

1.range-the range of variables, that is, the range of curves: abscissa; The ordinate.

Methods: ① Observation; ② Algebraic method.

2. Symmetry-it is not only an axisymmetric figure, but also an axisymmetric figure; It is also a central symmetric figure.

Methods: ① Observation; ② Definition method.

3. Vertex: the major axis of the ellipse, the minor axis of the ellipse,

The intersection of an ellipse and four symmetrical axes is called the vertex of an ellipse.

4. Eccentricity: describes the flatness of the ellipse. The ratio of the focal point of an ellipse to the length of its long axis is called eccentricity.

It can be understood as the degree to which the two focal points leave the center under the premise that the length of the long axis of the ellipse remains unchanged.

Step 5: Example

Example 4 Find the fixed-point coordinates of the length, eccentricity, focus, major axis and minor axis of an ellipse.

Tip: turn the general equation into a standard equation.

(Student answers-teacher writes)

Exercise: Find the length of major axis and minor axis, eccentricity, focal point coordinates and fixed point coordinates of ellipse and ellipse.

(Student Performance Committee-Teacher's Comments)

The ratio of the distance from a point to a fixed point to the distance from a straight line is constant, so find the trajectory of the point.

(Teacher Analysis-Demonstration Writing)

Third, classroom exercises:

① Compare the shapes of the following groups of ellipses, which is rounder and which is flatter?

(1) and (2) and (students answer and give reasons)

② Find the standard equation of ellipse suitable for the following conditions.

(1) passing point

(2) The length of the major axis is twice that of the minor axis, and it passes through this point.

(3) The focal length is, and the eccentricity is equal to.

(Student performance board, teacher comments)

③ Homework: Question 4.