## The Parabola and the Hyperbola

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 shows how the conic sections, the parabola and the hyperbola are formed.  Parabola Hyperbola (plane is parallel to side of cone) (plane is steeper than side of cone)

### The Basic Parabola

The basic equation of a parabola, is given by

 y2 = 4ax

where a is constant.

 The formal definition of a parabola is given in terms of a line called the directrix and a point called the focus. The parabola is defined as being the locus of a point which moves so that it is always equidistant from the focus point and the directrix line. Using the equation y2 = 4ax, the focus is at the point (a, 0) and the directrix is the line x = -a. a is called the focal length Also the point (0, 0) has a special name. It is called the vertex. ### The General Equation of a Parabola

As with circles and ellipses the method of translation can be used to find parabolas with a vertex at a point other than (0, 0)

The general equation is:

 (y − k)2 = 4a(x − h)

Where the vertex of the parabola is (h, k).

 Example For the parabola (y + 2)2 = 12(x − 10) Vertex is at (10, -2) Directrix is at x = 7 The constant a has a value 3 (since 4a = 12), so the focus is at (13, -2) ### The Basic Hyperbola

The hyperbolae studied previously were called rectangular hyperbolae because the asymptotes were at right angles.

The basic equation of a hyperbola is: The x-intercepts are (-a, 0) and (a, 0).

The equations of the asymtotes are The centre of the hyperbola is where the asymptotes cross.

 Example For the hyperbola By letting y = 0 The x-intercepts are x = ± 3 The two asymptotes are:  ### The General Equation of an Hyperbola

For hyperbolas with centres different from (0, 0) the equation is changed to: Where the centre of the hyperbola is (h, k).

 Example For the hyperbola The centre is (-3, 1) The two asymptotes are:  