Mathematics Lab Activity-18 Class XI

Mathematics Laboratory Activities on Conic Sections for class XI students Non-Medical. These activities are according to the CBSE syllabus

Chapter - 10 conic sections
Activity - 18

Objective

To construct an ellipse when two fixed points are given.

Material Required

Rectangular cardboard, coloured chart paper on a rectangular cardboard of a convenient size.

Theory

An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is constant.

Foci: Two fixed points are called the foci of the ellipse.

Centre: Mid- point of the line segment joining the foci of the ellipse is called its centre.

Major Axis: Line segment through the foci of the ellipse is called its major axis. Length of major axis is 2a.

Minor Axis: A line segment through the centre of the ellipse and is perpendicular to the major axis is called the minor axis. Length of minor axis is 2b.

Equation of the ellipse with centre at the origin and major axis is along the x- axis is  .

Procedure

1. Paste a coloured chart paper on a rectangular cardboard of convenient size.

2. Draw a horizontal line on the chart paper.

3. Mark two fixed points F₁ and F₂ on this line such that the distance between them is 8cm.

4. Fix two nails at the points  F₁, F₂.

5.Take a string whose length (say 14 cm) is more than the distance (7 cm ) between the two fixed points. F₁ and F₂

6. Fix the two ends of the string at the two nails at F₁ and F₂ .

7. Stretch the string in the form of a loop without slack with the help of a pencil and mark at least 10 points P₁, P₂, P₃, ………………,on each side of the line segment joining the points F₁ and F₂ as shown in figure 26.1.

8. Join all the points   P₁, P₂, P₃, ………………, P₂₀ to form an ellipse

Observations

By actual measurement, we find that :

1. P₁ F₁ + P₁ F₂ =2cm + 9 cm = 11 cm = K₁

2. P₂ F₁ + P₂ F₂ = 3cm + 8cm = 11 cm = K₂

3. P₃ F₁ + P₃ F₂ = 4cm + 7cm = 11 cm =  K₃

4. P₄ F₁ + P₄ F₂ = 5cm + 6cm = 11 cm =  K₄

5. P₅ F₁ + P₅ F₂ = 6cm + 5cm = 11 cm =  K₅

…………………………………………………………………………….

…………………………………………………………………………….

It will be found that . K₁ = K₂ = K₃ = K₄ ………. = K₂₀ = 11 cm

It leads to the conclusion that the sum of the distances of the points P₁, P₂, P₃,  ………. From the two fixed points F₁ and F₂  is the same.

This is imply that  the curve obtained is an ellipse.

Result

We can construct an ellipse when two fixed points are given. 

Applications

This activity can be used to explain the property of an ellipse that the sum of the distances of any point on the ellipse from its two foci is constant and is equal to length of its major axis.

VIVA – VOICE

Q. 1.  What are the two fixed points of the ellipse called?

Ans. Foci.

Q.2. What is an ellipse?

Ans. An ellipse is the set of all points in a plane, the sum of whose distances from two fixed points in the plane is constant.

Q.3. What is the equation of an ellipse with foci on the x-axis?

Ans.  

Q.4. Define eccentricity of an ellipse?

Ans. The eccentricity of an ellipse is the ratio of the distances from the centre of the ellipse to one of the foci and to one of  the vertices of the ellipse. It is denoted by e.

Q.5. Define latus rectum of an ellipse?

Ans. Latus rectum of an ellipse is a line segment perpendicular to the major axis through any one of the foci and whose end points lie on the ellipse.

Q.6. What is the length of the latus rectum of the ellipse   

Ans.  

Q.7. Define centre of an ellipse?

Ans. The mid-point of the line segment joining the foci is called centre of ellipse.

Q.8. Define major axis of an ellipse ?

Ans. The line segment through the foci of an ellipse is called major axis of  an ellipse.

Q.9. Define minor axis of an ellipse ?

Ans. The line segment through the centre and perpendicular to the major axis is called the minor axis.

Q.10. What do you mean by vertices of an ellipse?

Ans. The end points of the major axis are called as vertices of an ellipse.

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