Equation of a circle in different forms explained in detail

Do you know that any circle can be represented by an equation? Now we will learn the equation of a circle and its various forms. You should understand its importance in exams. If you are attempting any competitive exams then definitely you have to learn about circle equations. Not only equations we have some other concepts on the circle. But right now we will concentrate on equations of the circle.  Remaining topics associated with a circle I will discuss later.

Equation of a circle

We know that a straight line can be represented by a linear equation. And also we studied different forms straight line equations in coordinate geometry. In the same way, we can represent a circle by an equation. Like straight lines, a circle equation can be also represented in different forms. Now we will list out all the equations one by one.

The Equation of a circle in various forms:

»General form:

X2 + y2 + 2gx + 2fy + c = 0 is equation of circle with centre (-g, -f) and radius r = square root of g2 + f2 – c

»Diameter form:

Let us consider a circle with centre C and let P ( x1 , y1) and Q ( x2 , y2) be any two points on a circle. Then the equation of a circle is given as

  (y – y1) (y – y2) + (x – x1) (x – x2) = 0

»Centre – radius form:


The centre of a circle be (a, b) and radius of the circle be represented by the letter “r” then the equation of a circle is given by

(x-a) 2 + (y-b) 2 = r

Equation of a circle

»Parametric form:

Let us consider the circle with centre C (a, b), radius r and let α be the parameter. Then the coordinates of any point on the circle will be of the form (a + r cosα, b + r sinα ).

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