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.
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Parabola
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Hyperbola
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(plane is parallel to side of cone)
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(plane is steeper than side of cone)
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The Basic Parabola
The basic equation of a parabola, is given by
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y2 = 4ax
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where a is constant.
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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 is called the focal length Also the point (0, 0) has a special name. It is called the vertex. |
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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:
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(y − k)2 = 4a(x − h)
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Where the vertex of the parabola is (h, k).
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Example For the parabola (y + 2)2 = 12(x − 10)
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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:
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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.
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Example For the hyperbola
The two asymptotes are: |
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The General Equation of an Hyperbola
For hyperbolas with centres different from (0, 0) the equation is changed to:
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Where the centre of the hyperbola is (h, k).
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Example For the hyperbola The centre is (-3, 1) |
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