Tuesday, August 26, 2008

Ellipse - Definitions - July Dec Revision

1. An ellipse is the locus of a point which moves in a plane such that its distance from a fixed point (the focus) is always 'e' times its distance from a fixed line (the directrix), where e<1 is the eccentricity.

le the focus be at S(ae,0) and the directrix be x = a/e.

Then the equations of the ellipse is

x²/a² + y² /[a²(1-e²)] = 1

As e<1, 1-e²>0, we can write a² = b²/(1-e²) or

b² = a²(1-e²)

x²/a² + y² /b² = 1

The piont (0,0) is the centre of the ellipse.

The major axis will be along x axis and its length is 2a

The minor axis will be along y axis and its length is 2b.

The point (a,0) and (-a,0) are the two vertices of the ellipse.

A line through the focus of the ellipse perpendicular to the its major axis is called its latus recturm. Its equation is x = ae and its length is 2b²/a.

If we take (-ae,0) as the focus and the line x = -a/e as the directrix, we get the same equation of the ellipse. Therefore ellipse has two foci, (ae,0) and (-ae,0) and it has two directrices, x = a/e and x = -a/e respectively. It has also two latera recta whose equations are, x = ae and x = -ae.

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