Construction of Parabola by Eccentricity Method

CONSTRUCTION OF PARABOLA BY ECCENTRICITY METHOD:

Data known : Distance of the Focus from Directrix, and eccentricity e may be taken as 

1. Points to understand:

Understanding the following points will be very much useful to construct a Parabola by eccentricity method.

1. A parabola is an open curve and unlimited.

2. A parabola has only one focus and one directrix.

3. Eccentricity of parabola is 1 (unity) ie., the distance of any point on parabola from focus is equal to the distance of the point, measured from directrix.

4. Vertex of parabola is the mid point of the distance of Focus measured from directrix.

Example 6: Draw a parabola when the distance between focus and the directrix is 50 mm. Also draw a tangent and a normal at a point 60 mm from the directrix.

Construction:

1. Draw the Directrix DD'.

2. Mark a point A on DD' and draw the axis AA' through A and perpendicular to DD'.

3. Mark the Focus point F on the axis such that AF = 50 mm (given distance)

4. Mark vertex of Parabola V, such that V is the mid point of AF (since = AV / VF = 1)

5. Mark number of points 1, 2, 3 etc., from V at approximately equal intervals (need not be equal) along the axis AA' and erect vertical lines through these points. 

6. With the Focus F as centre and radius equal to A-1, draw two arcs on either side of the axis to intersect the vertical line drawn through the point 1 at P1 and P1'.

(Note that FP1 = FP1 = A - 1). 

7. Similarly obtain P2 and P2', P3 and P3' …. etc., by drawing arcs with F as centre and A-2, A-3 as radius ... etc., to intersect perpendicular lines drawn through the points 2, 3 ...... etc.) 

8. Join the points.... P2', P1', V, P1, P2 … etc., by a smooth parabolic curve.

To draw Tangent and Normal at any point on a Parabola

i) Mark a point P on the parabola at the given distance 60 mm from the directrix.

ii) Join P and Focus F.

iii) Draw a line FT perpendicular to PF meeting directrix at T.

iv) Join TP and extend, which is the required tangent, TT'.

v) Through the point P, draw a line NN' perpendicular to the tangent TT', which is the normal at P.