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.
 |
 |
|
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
|
|
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
|
|
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:
 |
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
|
 |
The General Equation of an Ellipse
For ellipses with centres different from (0, 0) the equation is changed to:
 |
Where the centre of the ellipse is (h, k) and the axes are a and b.
|
Example For the ellipse
|
|
