Flashcards: Circle

 Page 1


 
 
 
 
CIRCLE 
Standard Equation of Circle 
A circle is the locus of a point which moves such that its distance from a fixed point, called as centre, is 
equal to a given distance, called as radius. Thus, equation to a circle with centre ( ?? , ?? ) and radius ?? is : 
( ?? - ?? )
2
+ ( ?? - ?? )
2
= ?? 2
 
If the centre is origin then its equation reduces to ?? 2
+ ?? 2
= ?? 2
 
  
Page 2


 
 
 
 
CIRCLE 
Standard Equation of Circle 
A circle is the locus of a point which moves such that its distance from a fixed point, called as centre, is 
equal to a given distance, called as radius. Thus, equation to a circle with centre ( ?? , ?? ) and radius ?? is : 
( ?? - ?? )
2
+ ( ?? - ?? )
2
= ?? 2
 
If the centre is origin then its equation reduces to ?? 2
+ ?? 2
= ?? 2
 
  
 
 
 
 
 
General Equation of Circle 
The general equation of a circle is : 
?? = ?? 2
+ ?? 2
+ 2 ???? + 2 ???? + ?? = 0 
It has three arbitrary constants. Its centre is ( - ?? , - ?? ) and radius is v ?? 2
+ ?? 2
- ?? . The circle is real, point 
circle or imaginary according as ?? 2
+ ?? 2
- ?? > 0 , = 0, or < 0 (We are concerned to real circles only. 
Actually the concept of imaginary circles is beyond our scope.) ?? = 0 if circle passes through origin. 
  
Page 3


 
 
 
 
CIRCLE 
Standard Equation of Circle 
A circle is the locus of a point which moves such that its distance from a fixed point, called as centre, is 
equal to a given distance, called as radius. Thus, equation to a circle with centre ( ?? , ?? ) and radius ?? is : 
( ?? - ?? )
2
+ ( ?? - ?? )
2
= ?? 2
 
If the centre is origin then its equation reduces to ?? 2
+ ?? 2
= ?? 2
 
  
 
 
 
 
 
General Equation of Circle 
The general equation of a circle is : 
?? = ?? 2
+ ?? 2
+ 2 ???? + 2 ???? + ?? = 0 
It has three arbitrary constants. Its centre is ( - ?? , - ?? ) and radius is v ?? 2
+ ?? 2
- ?? . The circle is real, point 
circle or imaginary according as ?? 2
+ ?? 2
- ?? > 0 , = 0, or < 0 (We are concerned to real circles only. 
Actually the concept of imaginary circles is beyond our scope.) ?? = 0 if circle passes through origin. 
  
 
 
 
 
Other forms of the circle are as following: 
(i) General Equation of Second Degree ?? ?? 2
+ 2 h ???? + ?? ?? 2
+ 2 ???? + 2 ???? + ?? = 0 shall represent a circle if 
Coefficient of ?? 2
& ?? 2
 are equal i.e. ?? = ?? ? 0 & Coefficient of ???? is zero i.e. h = 0 and ?? 2
+ ?? 2
- ???? = 0 
  
Page 4


 
 
 
 
CIRCLE 
Standard Equation of Circle 
A circle is the locus of a point which moves such that its distance from a fixed point, called as centre, is 
equal to a given distance, called as radius. Thus, equation to a circle with centre ( ?? , ?? ) and radius ?? is : 
( ?? - ?? )
2
+ ( ?? - ?? )
2
= ?? 2
 
If the centre is origin then its equation reduces to ?? 2
+ ?? 2
= ?? 2
 
  
 
 
 
 
 
General Equation of Circle 
The general equation of a circle is : 
?? = ?? 2
+ ?? 2
+ 2 ???? + 2 ???? + ?? = 0 
It has three arbitrary constants. Its centre is ( - ?? , - ?? ) and radius is v ?? 2
+ ?? 2
- ?? . The circle is real, point 
circle or imaginary according as ?? 2
+ ?? 2
- ?? > 0 , = 0, or < 0 (We are concerned to real circles only. 
Actually the concept of imaginary circles is beyond our scope.) ?? = 0 if circle passes through origin. 
  
 
 
 
 
Other forms of the circle are as following: 
(i) General Equation of Second Degree ?? ?? 2
+ 2 h ???? + ?? ?? 2
+ 2 ???? + 2 ???? + ?? = 0 shall represent a circle if 
Coefficient of ?? 2
& ?? 2
 are equal i.e. ?? = ?? ? 0 & Coefficient of ???? is zero i.e. h = 0 and ?? 2
+ ?? 2
- ???? = 0 
  
 
 
 
Other forms of the circle are as following: 
(ii) Parametric Form 
Circle with centre ( ?? , ?? ) and radius ?? is ( ?? - ?? )
2
+ ( ?? - ?? )
2
= ?? 2
 
It can be represented in parametric form as 
?? = ?? + ?? c o s ? ?? ?? = ?? + ?? sin ? ?? ]
?? is parameter 
0 = ?? = 2 ?? 
A circle with centre ( 0 , 0 ) and radius ?? is 
?? = ?? c o s ? ?? ?? = ?? sin ? ?? ] ? ?? is parameter  
  
Page 5


 
 
 
 
CIRCLE 
Standard Equation of Circle 
A circle is the locus of a point which moves such that its distance from a fixed point, called as centre, is 
equal to a given distance, called as radius. Thus, equation to a circle with centre ( ?? , ?? ) and radius ?? is : 
( ?? - ?? )
2
+ ( ?? - ?? )
2
= ?? 2
 
If the centre is origin then its equation reduces to ?? 2
+ ?? 2
= ?? 2
 
  
 
 
 
 
 
General Equation of Circle 
The general equation of a circle is : 
?? = ?? 2
+ ?? 2
+ 2 ???? + 2 ???? + ?? = 0 
It has three arbitrary constants. Its centre is ( - ?? , - ?? ) and radius is v ?? 2
+ ?? 2
- ?? . The circle is real, point 
circle or imaginary according as ?? 2
+ ?? 2
- ?? > 0 , = 0, or < 0 (We are concerned to real circles only. 
Actually the concept of imaginary circles is beyond our scope.) ?? = 0 if circle passes through origin. 
  
 
 
 
 
Other forms of the circle are as following: 
(i) General Equation of Second Degree ?? ?? 2
+ 2 h ???? + ?? ?? 2
+ 2 ???? + 2 ???? + ?? = 0 shall represent a circle if 
Coefficient of ?? 2
& ?? 2
 are equal i.e. ?? = ?? ? 0 & Coefficient of ???? is zero i.e. h = 0 and ?? 2
+ ?? 2
- ???? = 0 
  
 
 
 
Other forms of the circle are as following: 
(ii) Parametric Form 
Circle with centre ( ?? , ?? ) and radius ?? is ( ?? - ?? )
2
+ ( ?? - ?? )
2
= ?? 2
 
It can be represented in parametric form as 
?? = ?? + ?? c o s ? ?? ?? = ?? + ?? sin ? ?? ]
?? is parameter 
0 = ?? = 2 ?? 
A circle with centre ( 0 , 0 ) and radius ?? is 
?? = ?? c o s ? ?? ?? = ?? sin ? ?? ] ? ?? is parameter  
  
 
 
 
 
 
(iii) Equation of Circle in Diametric Form