Mathematics Lab Activity-17 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 - 17

Objective

To constructing a parabola by using an alternative method.

Material Required

Cardboard, White paper, thread, ruler, nails, compass, pencil, sketch pens, adhesive, etc.

Theory

A conic is a locus of a point in a plane which moves so that its distance from a fixed point is in a constant ratio to its distance from a fixed straight line. The constant ratio is defined as eccentricity and is denoted by e

If e = 0, then the conic section is called a circle

If e = 1, then the conic is called a parabola.

If e < 1, then the conic is called ellipse.

If e > 1, then the conic is called a hyperbola.

Parabola:- A parabola is the set of all points in a plane equidistant from a fixed point and a fixed line in that plane. The fixed point is called the focus and the fixed line is called the Directrix.

Procedure

1. Take a card board of a convenient size and paste a white sheet on it.

2. Draw two mutually perpendicular lines  and m on the white paper fixed on the cardboard from any point .

3. Take any point S on the line m drawn on the white paper at a distance k units from the line as shown in fig.17.1.

Figure 17.1

4. Bisect OS at the point V i.e. V is the mid point of OS.

5. Mark points P₁, P₂, P₃, P₄  at equal distance  on VS and draw perpendiculars to the line m through these points as shown in figure.

6. With S as a centre and radius equal to OP₁, draw an arc cutting the perpendicular through  P₁ at the point A₁ and A₁’.

Similarly with S as a centre OP₂ as radius draw an arc cutting the perpendicular through   at the points A₂ and A₂’

 7. Repeat this process for some more points and  P₃ , P₄ ,…… and obtain points A₃ and A₃’ , A₄ and A₄’ ………….

8. Fix nails at these points i.e. at the points A₁, A₂, A₃,…… ; A₁’, A₂’, A₃’,……

9. Join the feet of the nails by a thread to get a curve as shown in the figure.

10. Distance of the point  A₁ from the line l = OP₁ = SA₁

11. Distance of the point  A₂ from the line l = OP₂ = SA₂

12. Distance of the point  A₃ from the line l = OP₃ = SA₃

13. Distance of the point A₁’ from the line l = OP₁ = SA₁’

14. Distance of the point A₂’ from the line l = OP₂ = SA₂’

15. Distance of the point A₃’ from the line l = OP₃ = SA₃’   And so on….

Therefore, every point on the curve is equidistant from the line l and the point  S . Therefore, the curve is a parabola. Whose focus is S and Directrix is the line l

Observations

By the actual measurement, we find that :

1. Distance of the point  A₁ from the line l = 3 cm ; SA₁ = 3 cm.

2. Distance of the point  A₂ from the line l = 4 cm ; SA₂ = 4 cm.

3. Distance of the point  A₃ from the line l = 5 cm ; SA₃ = 5 cm.

4. Distance of the point  A₄ from the line l =  6 cm ; SA₄ = 6 cm.

1. Distance of the point  A₁’ from the line l = 3 cm ; SA₁’ = 3 cm.

2. Distance of the point  A₂’ from the line l = 4 cm ; SA₂’ = 4 cm.

3. Distance of the point  A₃’ from the line l = 5 cm ; SA₃’ = 5 cm.

4. Distance of the point  A₄’ from the line l =  6 cm ; SA₄’ = 6 cm.

Clearly the distance of any point on the curve from the line l = Distance of the point from S.

Thus, the curve is a parabola with Directrix as the line l and focus as the point S.

Result

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. 

Applications

This activity is used to understand the terms related to a parabola, such as Directrix, vertex and focus of the parabola.

VIVA – VOICE

Q. 1. Who gave the name parabola and hyperbola?

Ans.  Apollonius

Q.2. What does a parabola means?

Ans. Parabola= Para + Bola       

‘Para’ means ‘For’ And ‘Bola’ means ‘throwing’

Therefore parabola is a shape described when we throw a ball in air.

Q.3.  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.4 What do you mean by axis of a parabola?

Ans. A line through the focus and perpendicular to the Directrix is called axis of the parabola.

Q.5. What is the vertex of a parabola?

Ans. The point of intersection of parabola and its axis is called vertex of parabola.

Q.6.  What is the length of the latus rectum of the parabola     ?

Ans. 4a

Q.7. Is parabola symmetric with respect to the axis of the parabola?

Ans. Yes.

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