Equation of a Circle

The equation of circle is actually an algebraic way to describe a circle, given the center and the length of the radius of the given circle. The equation of a circle is little bit different from the formulas that are used to calculate the area of circle or the circumference of a circle. This equation is mainly used in many problems of circles in coordinate geometry.

If we want to represent a circle on the Cartesian plane, then firstly we require the equation of the circle. We can draw a circle on a piece of paper, if we know its center of the circle and the length of its radius. In the same way, on a Cartesian plane, we can also draw a circle if we already know the coordinates of the center and its radius. A circle can be represented in many different forms, which are given below:

• General form
• Standard form
• Parametric form
• Polar form

In this article, we will learn about the equation of the circle, its various forms with graphs and solved examples.

Equation of Circle

An equation of a circle represents the actual position of a circle in a Cartesian plane. If we already know the coordinates of the center of the circle and the length of its radius, then it will be very easy to write the equation of a circle. The equation of circle actually represents all the points that lie on the circumference of the given circle.

A circle mainly represents the locus of points whose distance from a fixed point is a constant value. This fixed point is actually called the center of the circle and the constant value will be the radius r of the given circle. The standard equation of a circle with center at (x1,y1) and radius r is (x-x1)2+(y-y1)2=r2.

Different Forms of Equation of Circle

An equation of circle mainly represents the position of a circle on a cartesian plane. A circle can always be drawn on a piece of paper, if its center and the length of its radius is given. By using the equation of circle, we will be always able to find the coordinates of the center of the circle and its radius, we will also be able to draw the circle on the cartesian plane. There are different forms to represent the equation of a circle are:

• General form
• Standard form
• Parametric form
• Polar form

Let us learn the two common forms of the equation of circle that is – general form and standard form of the equation of circle. 

General Equation of a Circle

Actually, the general form of the equation of a circle is: x2 + y2+ 2gx + 2fy + c = 0. This general form is actually used to find the coordinates of the center of the circle and its radius, where g, f, c are always constants. The standard form which is very easy to understand, the general form of the equation of a circle makes it a little bit difficult to find any meaningful properties about the given circle. So, we will be using the completing the square formula to make a very fast conversion from the general form to the standard form.

Standard Equation of a Circle

The standard equation of a circle gives accurate information about the center of the circle and its radius and hence, it is very easy to read the center and the radius of the circle at a quick look. The standard equation of a circle with center at (x1,y1) and radius r is (x-x1)2+(y-y1)2=r2 where (x, y) is actually an arbitrary point on the circumference of the circle.

The distance between the given point on the circumference and the center is equal to the radius of the circle. Let us firstly try to apply the distance formula between these points.

(x-x1)2+(y-y1)2=r

By Squaring both sides, we will get actually the standard form of the equation of the circle, which is :-

x-x12+y-y12=r2

Parametric Equation of a Circle

We already know that the general form of the equation of a circle is x2 + y2+ 2hx + 2ky + c = 0.  We can also take a general point on the boundary of the circle, let us say (x, y). The line joining these general points and the center of the circle (-h, -k), actually makes an angle of θ. The parametric equation of circle can also be written as x2 + y2+ 2hx + 2ky + c = 0. Where x = -h + rcosθ and y = -k + rsinθ.

Polar Equation of a Circle

The polar form of the equation of the circle is almost very similar to that of the parametric form of the equation of the circle. We generally write the polar form of the equation of circle for the circle centered at the origin. Let us take a point P (rcosθ, rsinθ) on the boundary of the given circle, where r is actually the distance of the point from the origin. We may also know that the equation of circle centered at the origin and having radius ‘p’ is x 2+ y2= p2.

Let us substitute the value of x = rcosθ and y = rsinθ in the equation of the circle.

(rcosθ)2+ (rsinθ)2 = p2r2cos2θ +r2sin2θ = p2r2(cos2θ +sin2θ) = p2r2(1) = p2r = p

where p is actually the radius of the circle.

Important Notes on Equation of Circle

Here, I am giving a list of a few points that must be remembered while studying the equation of circle

• The general form of the equation of circle will always have x2+ y2 in the beginning.
• If a circle crosses both the axes, then there will be four points of intersection of the circle and the axes.
• If a circle touches both the axes, then there will always be only two points of contact.
• If the given equation is in the form x2+ y2+axy+C=0, then it is not any equation of the circle. Actually, there is no xy term in the equation of the circle.
• In polar form, the equation of circle is always representing or showing in the form of r and θ.
• Radius is the actual distance from the center to any point on the boundary or circumference of the circle. Therefore, the value of the radius of the circle is always positive.

Conclusion:

Based on above discussion, equation of circle can be put as follows:

1. An equation of circle mainly represents the position of a circle on a cartesian plane.
2. The general form is actually used to find the coordinates of the center of the circle and its radius.

While the standard equation of a circle gives accurate information about the center of the circle and its radius.