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Circle: Definition and Equation of a Circle

From bubbles to bangles to doughnuts to pizzas to car wheels there are so many things which are circular in shape. No wonder circles rule our world and command so much attention.

image

A circle is the set of all points which are at the same distance from a given point. But, since in mathematics every curve is represented by an expression/equation so is the case with circles also. Let's try to dig more into this.

Table of Contents

What is a Circle?

A circle can be defined as the locus of a point that moves in a plane such that its distance from a fixed point in that plane is always constant. 

The fixed point is known as the center of the circle and the constant distance is known as the radius of the circle.

Circle with center at  and radius

Equation of a Circle in Different Forms

  • Centre-radius form/Cartesian form/Standard form

The equation of the circle whose center is and radius ‘ is

Note:

  • By the equation of a circle, we mean the equation of circumference of the circle.
  • When the center of the circle coincides with the origin, i.e., and , the equation becomes . We call it the simplest form of a circle equation.
  • Diametric form

The equation of a circle whose end-points of a diameter are and is i.e  

Observe that the circle’s equation in diametric form is a combination of quadratic in having roots and quadratic in having roots .  

Circle with end points of diameter as  &

Practice Problems of Circle

Example : The circle passing through the point and touching the axis at also passes through the point

(a)  

(b)

(c)   

(d)

Solution : 

 

Step 1 :

Let the given circle be with centre and radius units.

Step 2 : 

passes through

Only the point satisfies the equation of the obtained circle, and hence it lies on the circle. Therefore, option (d) is the correct answer.

image2

 

 

Example : What is the image of the circle    in the line mirror ?

(a)

(b)

(c)

(d)

Solution :

 

Step 1 :

Let :

Centre of the circle

Radius     

Image of centre of with respect to the mirror l:
is the centre of circle and radius is same as that of

image4

 

Step 2 :

Let be the image of w.r.t line   

Where are the coefficients of and constant respectively and is the point whose image is to be found out.

Centre of

Radius

Equation of the required circle is

Hence, option (a) is the correct answer.

Example : If the abscissa and ordinates of two points and are the roots of the equation 

and , respectively, then find the equation of the circle with as the diameter.

Solution : 

Step 1 :

Let be roots of and , respectively.

,

Step 2 :

Equation of a circle with as the end points of the diameter is 

Therefore, the equation of the required circle is

Example : What is the equation of a circle that is passing through and and having the minimum possible radius?

(a)   

(b)     

(c)       

(d)

Solution : 

Step 1 :Let the equation of circle be

Since the circle passes through and , these points will satisfy its equation.

and

Solving and we get

The equation of circle becomes

Now the radius of the circle,

Step 2 : If is minimum then is minimum 

Also, the double derivative i.e. which is positive. Hence, is a point of minima

Hence the equation of the circle is

Hence, option (d) is the correct answer.

FAQs of Circle

Question1. Is a circle a two-dimensional figure?

Answer. Yes, a circle is a two dimensional shape.

Question2. If are two fixed points in a plane, then the locus of a point such that 

 is a straight line. Is the statement TRUE or FALSE?

Answer. False. The locus will be a circle for given conditions and will represent a straight line if

Question3. What is the circumference of a circle?

Answer. If we open a circle to form a straight line, then the length of the obtained line is the 

 circumference of the circle which is given by here is the radius of a circle.

Question4. What are congruent circles?

Answer. Two circles with the same radii are called congruent circles

Related Topics to Circles 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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