PURE MATHEMATICS - Coordinate Geometry

 

Circles & Ellipses

 

 

Circles

Ellipses

 

 

 

Circles

 

 

circle equation

 

 

Any point P is described by Pythagoras' Theorem.

 

So the equation of a circle with centre (0,0)and radius r is given by:

 

 

 

 

or in terms of parameters,

 

 

 

 

circle equation off-set from zero

 

 

For a circle with its centre off-set from the origin at a point C(a,b), again, by Pythagoras, the equation is given by:

 

 

 

 

Circle equation expanded(usual form)

 

 

circel equation - usual form

 

 

circel equation in final expanded form

 

 

 

Example

 

What is the radius and the coodinates of the centre of the circle with equation:

 

 

circle problem#1

 

 

circle problem#1 part b

 

 

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Ellipses

 

 

ellipse curve

 

 

 

The maximum displacement (a) along the x-axis is called the semi-major axis.

 

The maximum displacement along the y-axis (b) is called the semi-minor axis.

 

The names apply when a>b. When b>a the names are interchanged.

 

 

The Eccentricity (e) of an ellipse is defined as:

 

 

 

 

From the eccentricity we can define the points of focus (plural foci):

 

 

F1 (ae,0) and F2 (-ae,0)

 

 

and the directrices(directrix lines) at x = a/e , -a/e.

 

 

The directrices are two special lines parallel to the y-axis and either side of it.

 

(when the ellipse is centred at the origin)

 

Their unique property concerns the ratio of the distance between a point(P1) on the curve to a focal point(F1) and a line from the point to the directrix.

 

The ratio gives the eccentricity 'e' , where e < 1 :

 

 

 

 

 

 

 

 

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