Co-ordinate geometry of the circle (or simply ‘The Circle’) is the second of the co-ordinate topics on our course. It is therefore assumed that students studying The Circle have an adequate working knowledge of the ideas from The Line, the first of the co-ordinate geometry topics. For example, it is taken for granted that students are familiar with the perpendicular distance formula, which is frequently used in The Circle, in connection with questions on tangents. It is also assumed that students are familiar with terms such as tangent and chord, and can recall some of the more elementary geometry theorems from Junior Certificate, e.g., that a radius perpendicular to a chord bisects that chord.
Questions on The Circle generally occur at two levels. At the more straightforward or superficial level, we have to know formulae, learn to recognise the need for them, and substitute values. At the deeper level, information has to be analysed and often a diagram drawn. Then a strategy has to be devised, which may involve four or more separate stages. In such questions, we tend to employ methods rather than relying on formulae.
The study of Senior Cycle Co-ordinate Geometry of the Circle can be divided into the following sections:
1. Equation of a Circle
2. Two circles
3. Circle and Line
4. Two circles and a line
Equation of a Circle
A circle can be describe by giving the coordinates of its centre point and the length of its radius. The length of the radius can be given by Pythagoras’ theorem. If the circle is centred at the origin, (0,0), then the equation of the circle is given by x2 + y2 = r2 .
If we slide the circle away from the origin so that its centre is located at a point (h,k) we can apply the same reasoning to come up with the equation of a circle as given on page 19 of the F&T booklet. That is (x-h)2 + (y-k)2 = r2. This can be understood by examining the diagram below. This is the same as the equation for the length of a line segment given on page 18 of the F&T booklet. It just looks a bit different because they have used (x2-x1) on one page and (x-h) on the other, and in one give the formula in terms of the distance between the end points of the line segment and in the other in terms of the square of the length of the radius.
But r = √( x2+y2) is the same as x2+y2 = r2.
In other words the formula for a circle is just the distance formula applied to the radius of the circle, and both sides of the equation are squared just to get rid of the root sign.
The other version of the equation of a circle on page 19 of the booklet is
x2 + y2 + 2gx + 2fy + c = 0
Is obtained directly from (x-h)2 + (y-k)2 = r2 by renaming the centre coordinates (h,k) as (-g,-f), and expanding the left hand side to get
x2 + g2 + 2gx + y2 + f 2 + 2fy = r2.
Then bringing the r2 to the left hand side
x2 + g2 + 2gx + y2 + f 2 + 2fy – r2.= 0
and gathering all the constant number terms into a single value c to give
x2 + y2 + 2gx + 2fy + c = 0.
So c = g2 + f 2 – r2. Or r2. = g2 + f 2-c
resulting in the given formula for the radius r : r2. = g2 + f 2-c
or r. = √( g2 + f 2-c)