## The Circle and the Eclipse

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 Centre is (0, 0) Radius is √ 9 = 3 ### 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 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 Centre is (0, 0) semi-major axis is 4 semi-minor axis is 3 ### 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 Centre is (-3, 1) semi-major axis is 5 semi-minor axis is 4 