We know that we can find out about the measurements of properties of a given geometrical figure if we know the right formula to calculate it. Thus it is very important for us to know what would be the general equation of any given geometrical figure so that we may calculate its various measurements. Let us find out about the general properties and the general equation of a circle in this section.
What is a circle?
A circle is a simple bound round figure and can be drawn on a coordinate plane. It is a 2D figure and it can be drawn with a fixed point which is known as the centre and then the circumference is drawn with points which are all equidistant from the centre.
In order for us to arrive at the general equation of a circle, we need to first understand the basics that make up this shape.
Now take a look at this picture. The fixed point of origin of the circle, which is denoted by an O here, is the centre of the circle.
C here is the circumference of the circle which bounds the entire shape and every point on the circumference is equidistant from the centre. The circumference of a circle with a diameter D is represented as π*D. (π=22/7 or 3.141592…)
R is the radius of the circle which is the distance between the centre to any point on the circumference. The longest distance between two points on the circumference is defined as the diameter.
Diameter is the double of the radius in a circle. Thus, if radius is represented by R, then the diameter of that same circle will be represented as 2R.
The total area enclosed by the circle can be represented as πr^2 .
Simple properties of a circle
Apart from these basic properties, there are other properties of a circle too that one needs to know about. Let us take a look at some of them:
|Arc||A line which is curved and forms a part of the circle and is measured in radians. It can never be a full circle.|
|Chord||Any segment of line that touches two points on the circumference of the circle. The Diameter of a circle is its longest chord.|
|Tangent||A line that touches only one point on the circumference and runs perpendicularly to the centre of the circle|
|Sector||A certain section of the total area covered by the circle|
Understanding of these properties will help us to understand how we can arrive at the general equation of a circle.
Deduction of the general equation of a circle
One needs to be aware of the basic properties of a coordinate plane because otherwise, it is not possible to calculate the general equation of any given circle. So first we need to understand what a coordinate plane is.
A coordinate plane is basically formed with the help of two number lines and is two dimensional in nature.
Consider the two number lines that meet at the point mentioned as O, which is known as the Origin. This is not to be confused with the Origin or centre of a circle, which may or may not have its centre at the origin of the coordinate plane. Now the number line that runs vertically is known as the x axis and the number line that runs horizontally is known as the y axis. Now when we calculate the equation of a circle, we do so by placing it on the coordinate plane and hence it is important for us to have a basic idea of what a coordinate plane is.
There are mainly two scenarios when it comes to deducing the equation of a circle- i. When the centre of the circle is the origin point of the coordinate plane, and ii. When the centre of the circle is not the origin point of the coordinate plane.
- When the centre of the circle is the origin point of the coordinate plane:
Take any point on the circumference of the circle, say P which is graphically represented on the coordinate plane as x,y. Since O is the centre of the circle then OP will give us the radius which may be represented as a.
Since the distance between the point (x,y) from the centre, on a coordinate plane can be found with the help of a formula,
So, the equation of a circle within this case will be,
- When the centre of the circle is not the origin point of the coordinate plane
If h,k represents the centre C of the circle and x, y represent any point P on the circumference of the circle, then, in accordance with the formula of distance on coordinate planes,
Now, if the radius of the given circle be a, then the general equation of a circle with a different centre than the origin becomes,
Example: Suppose you have a circle whose centre is (4,5) and you know that the radius is 3 units. Then what will be the general equation of the circle?
Then, the general equation becomes,
(x−4)2+(y−5)2 = 32
x2−8x+16+y2−10y+25 = 9
x2+y2−8x−10y+16 = 0
As we can find the general equation of a circle with the provided coordinates of centre and radius, we can also decipher the centre and radius from the general equation provided with the help of the formula,
x2+y2+2gx+2fy+c = 0, where (-g, -f) represent the centre of the circle and the radius is given by a=g2+f2−c
Always remember the following
- If g2+f2 = c, then it tells us that it is a point circle, that is the centre and the origin are the same.
- If g2+f2 < c, then the circle has a real centre but the radius is imaginary and hence it is an imaginary circle.
- If g2+f2 > c¸then it is a real circle, with a real radius.
Once you remember the equation of a circle, it becomes easier to solve various geometric problems and hence it is important that you get your base cleared in this matter.