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SPECIAL CURVES LOCI

The locus of a point is the path traced by the point when moving in accordance with a definite rule. In other words, it is all the positions occupied by a moving point according to a given law.

The locus of points equidistant from two fixed points is a straight line while locus of points equidistant from a fixed point is called a circle.

CONIC SECTIONS

The conic sections are the shapes which result when a cone is cut from different positions. The conic sections are triangle, circle, ellipse, parabola and hyperbola.

(ATTACH THE DIAGRAM SHOWING THE CONIC SECTIONS)

SPECIAL CURVES 1

ELLIPSE

The ellipse is a plane figure bounded by a curved line called circumference. Its longest diameter is called major axis while its shortest diameter is the minor axis. The two axes bisect each other at right angles. The ellipse is symmetrical about the axes.

It can also be defined as the locus of points which moves such that the sum of the distances from two fixed points is constant.

The area of an ellipse is given by πab, where a is the longest radius and b is the shortest radius.

The solid formed by rotating an ellipse about an axis is called Ellipsoid. The eccentricity of an ellipse is less than unity or less than 1. The two fixed points in an ellipse are called foci (singular is focus).

METHODS OF CONSTRUCTING ELLIPSE

FOCI / INTERSECTING ARCS METHOD

Given

Major axis : 100mm

Minor axis : 60mm

Procedure

Draw the given axes AB and CD (100mm and 60mm respectively).

Locate the focal points using radius OA and centre at C.

Divide F1F2 into a number of equal parts e.g. 8

Centre at F1 and radii A1, A2, A3, A4, A5,etc draw arcs above and below.

Centre at F2, radii B1, B2, B3, B4, B5, etc intersect the previous arcs.

Join their points of intersection to get the ellipse.

RECTANGLE METHOD

Given

Major axis : 100mm

Minor axis : 60mm

Procedure

Draw the rectangle(ABCD) with AB = major axis =100mm and BC = minor axis = 60mm.

Locate the axes PQ and RS, they intersect at O.

Divide DP, AP, CQ and BQ into the same number of equal parts.

Divide PO and QO into the same number of equal parts,

Radiate lines from R to all divisions on DP and CQ.

Radiate lines from S to all divisions on AP and BQ.

Radiate lines from R passing through the divisions on OP and OQ to intersect with their corresponding radial lines from S.

Radiate lines from S passing through the division the divisions on OP and OQ to intersect with their corresponding radial lines from R.

Join their points of intersection to get the ellipse.

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