Coordinate Geometry at this level involves study of a selection of curves called the Conic Sections.

The Conic Sections

The conic sections are the circle, the ellipse, the parabola and the hyperbola.

These curves are obtained when a plane intersects a double cone.

The diagrams show how the conic sections, the circle and the ellipse are formed.

Y12_The_Circle_and_the_Ellipse_01.gif
Y12_The_Circle_and_the_Ellipse_02.gif
Circles
Ellipses
(plane is parallel to base of cone)
(plane is flatter than side of cone)

The Basic Circle

The basic equation of a circle, centre (0, 0) and radius r is:

x2 + y2 = r2

This relation is different from most of the others dealt with so far. It is not a function. Many x-values map to two y-values.

In some ways this makes it more difficult to deal with. i.e. Plotting points, differentiating.

Example

For the circle x2 + y2 = 9

Centre is (0, 0)

Radius is √ 9 = 3

Y12_The_Circle_and_the_Ellipse_03.gif

 


The General Equation of a Circle

For circles with centres different from (0, 0) the equation is changed to:

(x – a)2 + (y – b)2 = r2

Where the centre of the circle is (a, b) and the radius is r.

Example

For the circle (x − 2)2 + (y + 1)2 = 16

Centre is (2, -1)

Radius is √ 16 = 4

Y12_The_Circle_and_the_Ellipse_03.gif

 

Completing the Square

If the equation is given in a form such as x2 − 6x + y2 + 8y = 11 it is difficult to see what the radius and centre are.

The equation has to be re-written in the form (x – a)2 + (y – b)2 = r2.

To do this, both the x and y terms have the square completed.

For x2 − 6x to complete the square 9 has to be added to give x2 − 6x + 9 = (x − 3)2

For y2 + 8y to complete the square 16 has to be added to give x2 + 8x + 16 = (x + 4)2

Thus 9 + 16 have to be added to both sides to give

x2 − 6x + 9 + y2 + 8y +16 = 11 + 9 + 16

⇒ (x − 3)2 + (y + 4)2 = 36

which is the equation of a circle, centre (3, -4) with radius 6


 

The Basic Ellipse

The equation of the ellipse is closely related to that of the circle

The basic equation of an ellipse, centre (0, 0) is:

Y12_The_Circle_and_the_Ellipse_05.gif

where a is called the semi-major axis and b is called the semi-minor axis( if a is bigger than b.)

Example

For the ellipse Y12_The_Circle_and_the_Ellipse_06.gif

Centre is (0, 0)

semi-major axis is 4

semi-minor axis is 3

Y12_The_Circle_and_the_Ellipse_07.gif

The General Equation of an Ellipse

For ellipses with centres different from (0, 0) the equation is changed to:

Y12_The_Circle_and_the_Ellipse_08.gif

Where the centre of the ellipse is (h, k) and the axes are a and b.

Example

For the ellipse Y12_The_Circle_and_the_Ellipse_09.gif

Centre is (-3, 1)

semi-major axis is 5

semi-minor axis is 4

Y12_The_Circle_and_the_Ellipse_10.gif