To constructing a parabola by using an alternative method.
Cardboard, White paper, thread, ruler, nails, compass, pencil, sketch pens, adhesive, etc.
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.
1. Take a card board of a convenient size and paste a white sheet on it.
2. Draw two mutually
perpendicular lines
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.
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
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
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.
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.
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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