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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

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

image                     

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.   

image            

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 : 

 

Step 1 :

Let the circle be :

 

This circle passes through point .

 

 

This circle also passes through point

 

               

 

 

This circle also passes through point .

            

 

Step 2 :

 

Adding gives,

Substituting   in equation gives,

 

 

Substituting   and in equation gives,

 

 

Substituting  , and in equation ,

we get the equation of the circle :

 

 

 

Step 3 :

The circle formed by the three points 

is : , is also satisfied by point .

 

i.e., the points and lie on the same circle.Hence proved.

 

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

NCERT Class 11 Maths Chapters

Sets Relations and Functions Triginometric Functions
Mathematical Induction Numbers and Quadriatic Equations Linear Inequalities
Premutations and Combinations Binomial Theorem Sequence and Series
Straight Lines Conic Sections 3 D Geometry
Limits and Derivatives Mathematical Reasoning Statistics
Probability  
 
 
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