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:
x^{2} + y^{2} = r^{2}

This relation is different from most of the others dealt with so far. It is not a function. Many xvalues map to two yvalues.
In some ways this makes it more difficult to deal with. i.e. Plotting points, differentiating.
Example For the circle x^{2} + 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} = r^{2}

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 x^{2} − 6x + y^{2} + 8y = 11 it is difficult to see what the radius and centre are.
The equation has to be rewritten in the form (x – a)^{2} + (y – b)^{2} = r^{2}.
To do this, both the x and y terms have the square completed.
For x^{2} − 6x to complete the square 9 has to be added to give x^{2} − 6x + 9 = (x − 3)^{2}
For y^{2} + 8y to complete the square 16 has to be added to give x^{2} + 8x + 16 = (x + 4)^{2}
Thus 9 + 16 have to be added to both sides to give
x^{2} − 6x + 9 + y^{2} + 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 semimajor axis and b is called the semiminor 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

