Construction of Ellipse

Sample Problem 1:
a) Construct an ellipse when the distance between the focus and the directrix is 30mm and the eccentricity is 3/4.
b) Draw the tangent and normal at any point p on the curve using directrix.

Steps for Construction of Ellipse.

Solution: a)

Draw a vertical line DD to represent the directrix.At any point C on it draw a line perpendicular to the directrix to represent the axis CC’.
Distance between the focus and the directrix is 30 mm. Hence mark F1, the focus such that CF1=30mm.
e=3/4. So construct a right angled Δ CXY such that XY/CX = 3units/4units . (X is any point on the axis).
From F1 draw a 45º line to cut CY at S.From S erect Vertical to intersect CF1 at V1, the vertex. Now SV1 = F1V1.
ΔCXY is similar to ΔCV1S.
∴SV1/CV1 = XY/CX = F1V1/CV1 = 3/4.
From F1 draw another 45º line to intersect the extension of CY at T. From T erect vertical to intersect the axis at V2, another vertex. V1V2 = Major Axis.
Along the major axis, mark points 1,2,….,10 at approximately equal intervals. Through these points erect verticals to intersect CY (produced if necessary) at 1′,2′,…,10′.
With 11′ as radius and F1 as centre draw two arcs on either side of the axis to intersect the vertical line drawn through 1 at P1 and P1′ .
With 22′ as radius and F1 as centre draw two arcs on either side of the axis to intersect the vertical line drawn through 2 at P2 and P2′ .
Repeat the above and obtain P3 and P3′,….,P10 and P10′ corresponding to 2,3,….,10 respectively. Draw a smooth ellipse passing through V1,P1,…,P10,V2,P10′,…,P1′,V1.
To mark another Focus F2: Mark F2 on the axis such that V2F2 = V1F1.
To mark another Directrix D’D’ : Mark C’ along the axis such that C’V2 = CV1. Through C’ draw a vertical line D’D’.

Solution (b)

Mark the given point P and join PF1. At F1 draw a line perpendicular to PF1 to cut DD at Q. Join QP and extend it. QP is the tangent at P.
Through P, draw a line NM perpendicular to QP. NM is the normal at P.