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In addition to the revision notes for Circle Graphs on this page, you can also access the following Types of Graphs learning resources for Circle Graphs
| Tutorial ID | Title | Tutorial | Video Tutorial | Revision Notes | Revision Questions |
|---|---|---|---|---|---|
| 15.6 | Circle Graphs |
In these revision notes for Circle Graphs, we cover the following key points:
By definition, a circle is the set of all points that have the same distance from a fixed point called 'centre'.
We express the centre of a circle by the letter O (or sometimes C) and the common distance called radius by the letter R (or sometimes r). Given this description, it is clear that a radius has a single centre but an infinity number of radii, as each of them represents the shortest path that connects the centre and a given point of the circle.
Another important feature of circles is the diameter D, which represents the longest segment we can draw to connect two points of the circle. The diameter is twice the radius in length, i.e. D = 2r.
The set of points the definition refers to, is nothing more but a line - more precisely, a closed curve that surrounds the centre O (or C) of the circle. A circle includes only the points that belong to the closed line, not all points that are in the interior of the shape.
The equation of a circle is based on that of the distance between two points. This is because all points of the circle have the same distance from the origin, which is a fixed point. In this regard, one of the points is always the origin O and the other point can be any of the points in the circle. Therefore, since the equation of the distance between two points is
after raising it into the second power and replacing the distance d with the radius r, we obtain the general equation of the circle
where (a, b) are the coordinates of the centre C.
As a special case of the equation of a circle, we can mention situations where the centre of the circle corresponds to the origin O(0, 0). In these cases, the equation of circle becomes
or
As special cases, we can also mention those in which the centre C lies on any of the axes. Thus, if the centre C lies on the X-axis (yC = b = 0), the equation of the circle becomes
and if the centre C lies on the vertical axis (xC = a = 0), the equation of the circle becomes
Sometimes, the equation of a circle is given as a polynomial in the expanded form
where a, b, c and d are coefficients whilee is a constant.
Obviously, in circles the two coefficients a and b must be equal; otherwise, there is no symmetry around the centre C. If the value of these two coefficients is different from 1, then we multiply all terms of the corresponding polynomial by 1/a (or 1/b/ if you wish) in order to write it in the simplest form with a = b = 1.
Not all the second-degree polynomials with two variables represent circles in the coordinates system. Therefore, the first thing to do when dealing with such situations is to check whether the given second-degree polynomial with two variables can be expressed in the factorized form (x - a)2 + (y - b)2 = r2 or not. For this, we have to take into account the expansion of each of the individual monomials, which are of the form
We can check whether a second-degree polynomial with two variables represents a circle or not, we can group all terms according to the variable they represent and check whether it is possible to obtain two separate binomials - one for each variable - and a negative constant, which when sent to the right side becomes positive. This is because in the equation of a circle the constant shows the radius raised to the second power, so it must be positive when written on the right side (the square of a negative number does not exist in the set of the real numbers).
The procedure to find the equation of a circle when the graph is shown is as follows.
Step 1 - The first thing to do is to detect the four extreme points of the circle: the uppermost, the lowermost, the leftmost and the rightmost ones. In this way, we obtain two crossing segments that represent two perpendicular diameters.
Step 2 - Using the midpoint formula for each segment to identify the centre C. Obviously, we need to use twice this formula, one for the horizontal and the other for the vertical direction.
Step 3 - Calculate the radius by dividing the changing coordinate of any of the above diameters by 2.
Step 4 - Use the values found in the previous steps to write the equation of the unknown circle in the form (x - a)2 + (y - b)2 = r2.
We can find many more things than just the centre and the radius from the equation of a circle. For example, we can find the intercepts of the circle with any of the axes. There may be 0, 1 or 2 intercepts with each axis depending on the position and size of the circle. However, you don't need to draw a circle in the coordinates system to know the number of intercepts. You can simply substitute one of the coordinates by 0, and solve the remaining equation in terms of the other variable. This gives the intercept with the axis that contains this variable. If the circle touches an axis at a single point, we say it is tangent with that circle.
Another thing you can find using the equation of a circle is the equation of the horizontal or vertical diameter. Thus, if you know the centre C(a, b) and the radius r of a circle (both are easily identified if the equation of the circle is given), then the equation of the horizontal diameter is y = b, which corresponds to the vertical coordinate of the centre C. Likewise, the vertical diameter has the equation x = a, given that it must pass through the horizontal coordinate of the centre C.
As for the allowed values, it is obvious that the diameter must extend horizontally within the range a - r ≤ x ≤ a + r and vertically it must extend within the range b - r ≤ x ≤ b + r
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