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Parametric Equation: Parametric Equation of a Circle, Intercepts on the Axes and Solved Examples

The equation of a circle is what we already know but the equation of the same circle can be written in different ways, one of which is parametric form. 

While solving problems on circles it is often required to consider a general point which lies on the circle. Here both and are variables. Instead of considering two variables, parametric form of the circle helps us convert this general point into a single variable θ. So it is very helpful and we can solve problems easily by considering the parametric form of the circle. Let’s have a look at it to understand it in a better way. 

Table of Contents

  • Parametric equations of a circle
  • Intercepts on the axes
  • Practice Problems 
  • FAQs

Parametric Equations of a Circle

 

For simplest form of circle  

 

The parametric equations of the circle are ::

 

where


For the standard form of a circle with center at ( and radius  

 

The parametric equations of the circle are as follows :

 

and , where is the parameter and is equal to the angle between radius vector and .

()


For general form of circle and

 

The parametric equations of the circle are :

 

 


 

Concept Video

Parametric Equation of a Circle | MATHS Class 11/12 | JEE 2020/2021 | GB Sir

Intercepts on the Axes

Length of the intercept made by the circle

 

On axis is  
On axis is .

 

Note : General equation of the circle for different conditions:

 

  • If  

 

⇒ Circle intersects the axis at two distinct points


  • If  

 

⇒ Circle touches the axis


  • If  

 

⇒ Circle does not meet axis i.e completely lie above or below x-axis


  • If  

 

⇒ Circle intersects the axis at two distinct points


  • If  

 

⇒ Circle touches the axis 


  • If  

 

⇒ Circle does not meet axis


  • If  

 

⇒ Circle touches both the axes

 

 


 

Practice Problems of Parametric Equations

Example : For how many values of , does the circle and the coordinate axes have exactly three common points?

Answer: 

Given,  .

Comparing with the general equation of the circle, we get

The center of the circle is

 

Geometrically, the circle will have exactly three common points with coordinate axes in the following cases :

Case 1: 

 

Circle passing through origin i.e. (0,0)

 

 


Case 2 : 

 

Circle touching the axis and intersecting axis at two points 

and

∴   


Case 3 : 

 

Touching axis and intersecting axis at two points 

 and     

Hence, no value of is possible for this case.

 

 

 

 

Hence, only two values of is possible.

Example : Circle(s) touching the axis at a distance of units on the positive side from origin and having intercept of length on axis is/are as follows :     

(a)                              
(b)
 (c)                             
 (d)

Answer: 

 

Step 1 : 

 

The figure shows that the circle touching the axis at and having intercept

on the axis.

 

The abscissa of the center is .

Radius of the circle

 

      


Step 2 : 

 

∴ Centers of the circles are

 

∴ Equations of circles are 

 

 

Options (a),(c) are correct

Example : If the point holds good for , then find the maximum and minimum values of .

Answer: 

Step 1 :

Given, equation of the circle,

Comparing with ,

Center, and radius

Now, the parametric coordinates of a point on the given circle,

Step 2 :

Now,

We know that,

Therefore, , and

Example : Find the circumcenter of if , and .

Answer: 

Vertices of are given in the form of parametric coordinates for a circle that is , comparing with any one vertex of , we get

Therefore center of circle circumcentre of triangle

Hence, Circumcenter is

FAQ’s of Parametric Equations

Question 1.What is the radius vector of a circle?

Answer: line segment joining the center and a point on the circumference of a circle is called the radius vector of the circle. 

Question 2. Does the radius of the circle change by just shifting the center of the circle?

Answer: Since the radius is the distance between the center and circumference of the circle, it remains constant.

Question 3. Can a circle make an equal intercept on both axes?

Answer: Yes, if the value of and in the general form of the circle are equal, it will make an equal intercept on both the axes.

Question 4. What is the difference between the parametric form of a straight line and the parametric form of the circle? 

Answer: The parametric form of a circle is , Here ( is a parameter and is a constant whereas the parametric form of a straight line passing through is  , Here is a parameter and is the angle which the straight line makes with the positive direction of

 

Related Topics to Parametric 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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