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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.
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Circles
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Ellipses
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(plane is parallel to base of cone)
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(plane is flatter than side of cone)
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The Basic Circle
The basic equation of a circle, centre (0, 0) and radius r is:
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.
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Example
For the circle x2 + y2 = 9
Centre is (0, 0)
Radius is √ 9 = 3
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The General Equation of a Circle
For circles with centres different from (0, 0) the equation is changed
to:
Where the centre of the circle is (a, b) and the radius is r.
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Example
For the circle (x - 2)2 + (y + 1)2 = 16
Centre is (2, -1)
Radius is √ 16 = 4
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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.)
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Example
For the ellipse 
Centre is (0, 0)
semi-major axis is 4
semi-minor axis is 3
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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.
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Example
For the ellipse 
Centre is (-3, 1)
semi-major axis is 5
semi-minor axis is 4
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