Mathematics Lab Activity-16 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 - 16

Objective

To construct different type of conic sections.

Material Required

Transparent sheet, White paper, hard-board, scissors, adhesive.

Theory

The curves like parabola, ellipse and hyperbola are known as conic sections or simply conics because they can be obtain as a intersection of a plane with a double napped right circular cone.

CIRCLE:-  

When β(beta) = 90, then intersection of cone with plane is called circle. For circle the value of eccentricity = 0 or  e = 0

ELLIPSE:- 

When α < β < 90 then intersection of plane and cone gives an ellipse. For ellipse the value of eccentricity < 1 or  e < 1

PARABOLA:- 

When α= β then intersection of plane and cone gives parabola. For Parabola the value of eccentricity = 1 or  e = 1

HYPERBOLA:- 

When 0 < β < α then the intersection of plane and cone gives a hyperbola.

For Hyperbola the value of eccentricity > 1 or  e > 1

Procedure

1. Paste a white paper on a hardboard of a convenient size

2. Cut a transparent sheet in the shape of sector of a circle and fold it to obtain a right circular cone as shown in fig. 16.1.

Figure 16.1

3. Form four more such cones of the same size using the transparent sheet.

4. Put these cones on the hardboard.

5. Cut these cones with a transparent plane sheet in different positions as shown in fig. 16.2 to 16.5.

Figure 16.2                            Figure 16.3

Figure 16.4                                       Figure 16.5

6. When the transparent plane sheet cuts the cone in such a way that it (plane sheet) is parallel to the base (AB) of the cone, then the section so obtained is a circle as shown in fig. 16.2. Here e = 0.

7. When the transparent plane sheet cuts the cone in such a way that it (plane sheet) is inclined slightly to the axis (OC) of the cone, then the section so obtained is ellipse as shown in fig. 16.3. Here e < 1.

8. When the transparent plane sheet cuts the cone in such a way that it is parallel to the generator (slant height) DE of the cone, then the section so obtained is a parabola as shown in fig. 16.4. Here e = 1.

9. When the transparent plane sheet cuts the cone in such a way that it is parallel to the axis  of the cone, then the section so obtained is a hyperbola as shown in fig. 16.5. Here e > 1.

Observations

1. In fig 16.2, the transparent sheet is parallel to the base of the cone and the conic section so obtained is circle.

2. In fig 16.3, the plane sheet is inclined to the axis of the cone and the conic section so obtained is ellipse.

3. In fig 16.4, the plane sheet is parallel to the generator of the cone and the conic section so obtained is a parabola.

4. In fig 16.5, the plane sheet is parallel to the axis of the cone and the conic section so obtain is a part of a hyperbola.

Result

The various conic sections can be obtained as intersections of a plane with a double napped right circular cone.

Applications

This activity is very useful in understanding various types of conic sections. The curves have a very wide range of applications in real life situations and modern sciences such as planetary motion, design of telescope and antennas, reflectors in flashlight, reflection of light, beam of sound and automobile headlights etc.

VIVA – VOICE

Q. 1. What are conics?

Ans. The curves like a circle, an ellipse, a parabola and a hyperbola are the conic sections or simply conics .

Q.2 How can conics be obtained?

Ans.  Different type of conics can be obtained as a intersection of a plane with a double napped right circular cone .

Q.3 What is a circle?

Ans.  A circle is a set of all points in a plane that are equidistant from a fixed point in the plane.

Q.4 What is the equation of a circle with centre (h , k) and radius r

Ans.  

Q.5 What is an ellipse?

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

Q.6 What is an equation of ellipse with foci on the x-axis?

Ans.    

Q.7  What is a parabola?

Ans.  A parabola is a set of all points in a plane that are equidistant from a fixed line and a fixed point in the plane.

Q.8. What is an equation of parabola?

Ans. The equation of a parabola with focus at (a, 0) and Directrix x= -a is ;

Q.9. What is a hyperbola?

Ans.  A hyperbola is a set of all points in a plane, the difference of whose distance from two fixed points in the plane is constant.                           

Q.10. What is an equation of a hyperbola?

Ans. The equation of a hyperbola with foci on the x-axis is:

            

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