General equation of a circle

A circle can be described algebraically using its equation and the equation of a circle can be written in various forms.The one which is most commonly used is the General equation of a circle. Let us try to understand the general equation of circle in detail.

Table of Contents

• General Equation of a Circle
• Conversion of General Form of a Circle to Standard Form
• Circle Passing Through Three Non-Collinear Points
• Practice Problems on General Equation of a Circle 
• FAQ’s on General Equation of a Circle 

General Equation of a Circle

The equation represents the general equation of a circle with centre and radius

• If, then it is a real circle.
• If, then radius is zero and the circle is known as a point circle.
• If, then the radius is imaginary, and the circle is known as an imaginary circle with a real centre.

Note:The general equation of a second degree curve       will represent a circle if

• Coefficient of Coefficient of  
• The term containing is zero, i.e., coefficient of
• In general , For it represents a point circle.

Conversion of General Form of a Circle to Standard Form

 General equation of a circle

This is the required standard form of the type where

Circle Passing Through Three Non-Collinear Points

Let be the required circle passing through three non-collinear points , Then, 

We have 3 equations and 3 unknowns . On solving them, the values of can be obtained.

Practice Problems on General Equation of a Circle

Example : What is the equation of the circle described on the line segment joining the centers of the circles  and as a diameter?

(a)

(b)

(c)

(d)

Solution : 

Let and be the centre of the first and second circle respectively.On comparing the given equations of circles with the general equation of a circle,the centres

are obtained as: and

The equation of the circle having as the diameter is given by:

    where

Hence, option (a) is the correct answer.

Example : A circle of radius units touches the axes in the first quadrant. If the circle makes one complete roll on x-axis in the positive direction of x-axis, then what is the equation of the circle in the new position?

Solution : 

Let represent the circle in its original position and let represent the circle after rolling.

Centre of Let the centre of be .Let the circle touch the x-axis at in the original position while it touches the x-axis at after rolling.   

Now, Circumference of After rolling, the circle has moved a distance equal to the circumference and its radius is unchanged.

The equation of the circle is

Example :Find the equation of the circle that passes through the points and and whose centre lies on the line

Solution :

Let the equation of the circle be

The circle passes through and . On substituting the values respectively, we get,

- gives us

Given, Centre lies on

On solving , we get

.Now putting the values of in we obtain and

On substituting the values in equation , we obtain the equation of circle as

Example : Prove that the points and lie on the same circle.

Solution : 

FAQ’s on General Equation of a Circle

Question.1 If two lines and intersect coordinate axes at concyclic points then  Answer: What is the equation of the circle passing through these points?

The equation of circle passing through such concyclic points is , neglecting the term.

Question.2 What is the eccentricity of a circle?

Answer: Eccentricity basically denotes how un-cirular the curve is. Hence, the eccentricity of a circle is 0.

Question.3 What are concentric circles?

  Answer: Circles which are having same centre are called concentric circles

Question.4 What is the maximum number of circles that can pass through non-collinear points?

Answer: There is only one circle possible that can pass through non-collinear points.

Question.5 Is circle a function or not?

Answer: A circle is not a function as if you draw a vertical line ( parallel to axis ) then the line would intersect the circle in two distinct points means that the vertical line test fails.

Related Topics to Equation of a Circle in Maths

• Circles
• Particular Cases of a Circle
• Parametric Equations
• Tangent to a Circle

NCERT Class 11 Maths Chapters