,
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:

equation of a circle

or in terms of parameters,

parameter equation of a circle

 

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:

equation of a circle off-set from the origin

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, while the maximum displacement along the y-axis (b) is called the semi-minor axis. This is when a>b. When b>a the names are interchanged.

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

eccentricity equation

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 and x=-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' .

eccentricity ratio

 

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