Definition 
A circle is defned as the locus o a point which moves in a plane such that its distance rom a fxed point in that plane always remains the same i.e., constant.
 The fxed point is called the centre o the circle and the fxed distance is called the radius o the circle.
Standard forms of equation of a circle
(1) General equation of a circle :  The general
equation o a circle is !! !! =++++   cfy gx  y  x   where
g, f , c are constant.
(i) "entre o the circle is (#g, #f ). i.e., ( !
1−
Nature of the circle
(i) & !! >−+ cf g , then the radius o the circle will
'e real. ence, in this case, it is possi'le to draw a circle on a plane.
(ii) & !! =−+   cf g , then the radius o the circle
will 'e ero. *uch a circle is +nown as point circle.
(iii) & !! <−+   cf g , then the radius cf g   −+   !!
o the circle will 'e an imaginary num'er. ence, in this case, it is not possi'le to draw a circle.
The condition for the second degree equation to represent a circle :  The general equation
!! !   by hxy ax    ++   !!   =+++   cfy gx    represents a
circle i
(iv) !! ≥−+   acf g
(!) Central form of equation of a circle :  The equation o a circle having centre (h, k ) and radius r  is
!!! )()(   r k  y h x    =−+− & the centre is origin, then the equation o the circle is
!!! r  y  x    =+
(-) Circle on a given diameter : The equation o  the circle drawn on the straight line oining two given
points ),(   11  y  x    and
),(   !!  y  x   as diameter is
))(())(( !1!1   =−−+−−   y  y  y  y  x  x  x  x 
(/) Parametric co-ordinates
(i) The parametric co0ordinates o any point on the
circle !!! )()(   r k  y h x    =−+−   are given 'y
)sin,cos(   θ θ    r k r h   ++ , )!(   π θ  <≤ .
&n particular, co0ordinates o any point on the circle !!! r  y  x    =+  are )sin,cos(   θ  θ    r r  , )!(   π θ  <≤ .
(ii) The parametric co0ordinates o any point on the
circle !! !! =++++   cfy gx  y  x    are
θ cos)(   !! cf gg x    −++−=   and
θ sin)(   !! cf gf  y    −++−= , )!(   π θ  <≤
() Equation of a circle under given conditions
(2oving point)
Chapter
17
(i)  The equation o the circle through three non0
collinear points ),(),,(),,( --!!11   y  x C y  x B y  x  A
is
1
1
1
1
 y  x  y  x 
 y  x  y  x 
 y  x  y  x 
 y  x  y  x 
(ii) 4rom given three points ta+ing any two as extremities o diameter o a circle S 5 and equation o  straight line passing through these two points is L 5 .
 Then required equation o circle is =+   LS   λ  , where
λ   is a parameter, which can 'e ound out 'y putting
third point in the equation.
Equation of a circle in some special cases
(1) & centre o the circle is ),(   k h  and it passes
through origin then its equation is !!!!
)()(   k hk  y h x    +=−+− !!  y  x    +⇒
!!   =−−   ky hx  .
(!) & the circle touches  x 0axis then its equation is !!! )()(   k k  y h x    =±+± . (4our cases)
(-) & the circle touches  y 0axis then its equation is !!! )()(   hk  y h x    =±+± . (4our cases)
(/) & the circle touches 'oth the axes then its equation is
!!! )()(   r r  y r  x    =±+±  . (4our cases)
() & the circle touches  x- axis at origin then its equation is
!!! )(   k k  y  x    =±+   !!! =±+⇒   ky  y  x  . (Two
cases)
(6) & the circle touches  y-axis at origin, the
equation o circle is !!!)(   h y h x    =+± !!! =±+⇒   xh y  x  . (Two cases)
(7) & the circle passes through origin and cut intercepts a  and b  on axes, the equation o circle is
!! =−−+   by ax  y  x    and centre is
)!8,!8(   baC . (4our cases)
Intercepts on the axes
!! !! =++++   cfy gx  y  x    on  X   and Y   axes are
cg   − !!  and cf    −!!  respectively.
(i) The circle !! !! =++++   cfy gx  y  x   cuts the x-
axis in real and distinct points, touches or does not
meet in real points according as cg   <=> or,! .
(ii) *imilarly, the circle !! !! =++++   cfy gx  y  x 
cuts the  y-axis in real and distinct points, touches or does not meet in real points according as
cf    <=> or,! .
Position of a point with respect to a circle
A point ),(   11  y  x   lies outside, on or inside a circle
!! !! =++++≡   cfy gx  y  x S  according as
cfy gx  y  x S   ++++≡ 11 ! 1
! 11 !!   is positive, ero or
negative.
from a circle: 9et S 5 'e a circle and
),(   11  y  x  A  'e a point. & the diameter
o the circle through  A  is passing through the circle at P and Q, then
=−=   ::   r  AC AP   least distance;
=+=   r  AC AQ   greatest distance
where <r < is the radius and C is the centre o the circle.
Intersection of a line and a circle
(– h,k )k 
(–h,– k )
 The length o the intercept cut o rom the line
cmx  y    +=   'y the circle !!! a y  x    =+   is
!
!!!
two real and dierent points.
(ii) & )1( !!! mac   += , line will touch the circle.
  (iii) & )1( !!! <−+   cma , line will meet the circle
at two imaginary points.
(1) Point form
(i) The equation o tangent at ( x 1,  y 1) to circle !!! a y  x    =+  is !
11   a yy  xx    =+ .
at ),(   11  y  x   to circle
!! !! =++++   cfy gx  y  x  is
)()( 1111   =++++++   c y  y f  x  x g yy  xx  .
(!) Parametric form :  *ince parametric co0
ordinates o a point on the circle !!! a y  x    =+   is
),sin,cos(   θ  θ   aa   then equation o tangent at
)sin,cos(   θ  θ   aa  is !sin.cos.   aa y a x    =+   θ  θ    
or a y  x    =+   θ θ    sincos .
(-)  Slope form :  The straight line cmx  y    +=
     
   
 
+
±
9et PQ  and PR  'e two tangents drawn rom
),( 11  y  x P  to the circle .!!!! =++++   cfy gx  y  x 
 Then PQ 5 PR is called the length o tangent drawn rom point P  and is given 'y PQ 5PR
111 ! 1
Pair of tangents
4rom a given point ),(   11  y  x P  two tangents PQ and
PR  can 'e drawn to the circle
.!! !! =++++=   cfy gx  y  x S  
 Their com'ined equation is ! 1   T SS   = ,
where =S  is the equation o circle, =T   is the
equation o tangent at ),(   11  y  x   and S1 is o'tained 'y
replacing x   'y x 1 and y  'y y 1 in S.
Director circle
 The locus o the point o intersection o two perpendicular tangents to a circle is called the =irector circle.
9et the circle 'e !!! a y  x    =+ ,
then equation o director circle is !!!
!a y  x    =+ .
concentric circle whose radius is !  times the radius
o the given circle.
!!! !!!! =−−++++   f gcfy gx  y  x  .
Power of point with respect to a circle
9et ),(   11  y  x P   'e a point
outside the circle and PAB  and PC  drawn two secants. The
power o ),( 11  y  x P   with
respect to
is equal to PA . PB which is
111 ! 1
==∴   !
1)(.   SPBPA
*quare o the length o tangent. & P is outside, inside or on the circle then PA . PB is
?!e, #!e or ero respectively. @
ormal to a circle at a given point
 The normal o a circle at any point is a straight line, which is perpendicular to the tangent at the point and always passes through the centre o the circle.
(1) Equation of normal: The equation o normal to
the circle !! !! =++++   cfy gx  y  x    at any point
),(   11  y  x    is
1
1
1
1 .
 The equation o normal to the circle !!! a y  x    =+  at
any point ),(   11  y  x    is 11   =−   y  x  xy    or
11   y 
(!) Parametric form :  *ince parametric co0
ordinates o a point on the circle !!! a y  x    =+   is
)sin,cos(   θ  θ     aa .
1
∴   Dquation o normal at   )sin,cos(   θ  θ     aa   is
θ θ    sincos   a
 y  x  =  or θ tan x  y =   or mx  y =
where θ tan=m , which is slope orm o normal.
!hord of contact of tangents
(1) Chord of contact : The chord oining the points o contact o the two tangents to a conic drawn rom a given point, outside it, is called the chord o contact o  tangents.
(!) Equation of chord of contact :  The equation o the chord o contact o tangents drawn rom a point
),(   11  y  x   to the circle !!! a y  x    =+  is .!11   a yy  xx    =+
Dquation o chord o contact at ),(   11  y  x    to the
circle !! !! =++++   cfy gx  y  x    is
)()( 1111   =++++++   c y  y f  x  x g yy  xx  .
&t is clear rom a'ove that the equation to the chord o contact coincides with the equation o the tangent, i 
point ),(   11  y  x   lies on the circle.
 The length o chord o contact !!!   "r    −= ; ( "
'eing length o perpendicular rom centre to the chord)
Area o  APQ  is given 'y ! 1
! 1
+
−+ .
(-) Equation of the chord #isected at a given point :  The equation o the chord o the circle
!! !! =++++≡   cfy gx  y  x S   'isected at the point
),(   11  y  x   is given 'y 1ST  = .
i.e.,
fy gx  y  x c y  y f  x  x g yy  xx    +++=++++++ 11 ! 1
! 11111 !!)()(
!ommon chord of two circles
(1)  $e%nition :  The chord oining the points o  intersection o two given circles is called their common chord.
(!) Equation of common chord : The equation o  the common chord o two circles
  !! 111 !!
1   =++++≡   c y f  x g y  x S E.(i)
and !! !!! !!
E.(ii)
!1   =−SS .
! 1
=#C1  length o the perpendicular rom the centre
1C  to the common chord PQ.
Diameter of a circle  The locus o the middle points o a system o 
parallel chords o a circle is called a diameter o the circle.
 The equation o the diameter 'isecting parallel chords
cmx  y    +=  (c is a parameter) o 
the circle !!! a y  x    =+   is
.=+my  x 
Pole and Polar 
9et ),(   11  y  x P 'e any point inside or outside
the circle. =raw chords AB and A$ B$ passing through P. & tangents to the circle at A and B meet at Q (h, k ), then
locus o Q is called the polar o P with respect to circle and P is called the pole and i tangents to the circle at
 A$ and B$ meet at Q$, then the straight line QQ$ is polar with P as its pole.
Dquation o polar o the circle
!!!! =++++   cfy gx  y  x    %.r.&.  ),(   11  y  x    is
)()( 1111   =++++++   c y  y f  x  x g yy  xx  .
& the circle is !!! a y  x    =+ , then its polar %.r.&.
),(   11  y  x  is ! 11   =−+   a yy  xx  .
 The pole o the line =++   'my (x   with respect to
the circle !!! a y  x    =+ . 9et pole 'e ),,(   11  y  x    then
equation o polar with respect to the circle !!! a y  x    =+
is ! 11   =−+   a yy  xx  , which is same as =++   'my (x 
 Then '
a
m
 y 
   
  −−
'
Properties of pole and polar
(i) & the polar o ),(   11  y  x P  %.r.&. a circle passes
through ),(   !! y  x Q   then the polar o Q  will pass
through P  and such points are said to 'e conugate points.
(ii) & the pole o the line =++   cby ax    %.r.& . a
circle lies on another line ;111   =++   c y b x a  then the
pole o the second line will lie on the frst and such lines are said to 'e conugate lines.
!ommon tangents to two circles
=ierent cases o intersection o two circles G
9et the two circles 'e ! 1
! 1
E..(i)
and ! !
E..(ii)
with centres ),(   111   y  x C   and ),( !!!   y  x C   and
radii r 1  and r !  respectively. Then ollowing cases may arise G
Case ( : Fhen !1!1   ::   r r CC   +>  i.e., the distance
'etween the centres is greater than the sum o radii.
&n this case our common tangents can 'e drawn to the two circles, in which two are direct common tangents and the other two are transverse common tangents.
!
    
   
 
− −
− −
    
   
 
+ +
+ +
Case (( : Fhen !1!1   ::   r r CC   +=  i.e., the distance
'etween the centres is equal to the sum o radii.
&n this case two direct common tangents are real and distinct while the transverse tangents are coincident.
Case ((( :  Fhen !1!1   ::   r r CC   +<   i.e., the
distance 'etween the centres is less than sum o radii.
&n this case two direct common tangents are real and distinct while the transverse tangents are imaginary.
Case () :  Fhen ,::::   !1!1   r r CC   −=   i.e., the
distance 'etween the centres is equal to the dierence o the radii.
&n this case two tangents are real and coincident while the other two tangents are imaginary.
Case ) :  Fhen ,:::: !1!1   r r CC   −<   i.e., the
distance 'etween the centres is less than the dierence o the radii.
&n this case, all the our common tangents are imaginary.
"ngle of intersection of two circles
 The angle o intersection 'etween two circles S 5 and S< 5 is defned as the angle 'etween their tangents at their point o intersection.
& !! 111 !! =++++≡   c y f  x g y  x S
  !!< !!! !! =++++≡   c y f  x g y  x S
are two circles with radii !1,  r r    and +  'e the
distance 'etween their centres then the angle o 
intersection θ   'etween them is given 'y
!1
  * Circle and System of Circles
Condition of +rthogonality : & the angle o intersection o the two circles is
a right angle )B( ,=θ  , then
such circles are called orthogonal circles and condition or orthogonality is
!1!1!1 !!   ccf f gg   +=+ ,
#amil$ of circles
(1) The equation o the amily o circles passing through the point o intersection o two given circles S 5
and S$ 5 is given as <=+   SS   λ  , (where λ   is a
parameter, )1−≠λ 
(!) The equation o the amily o circles passing through the point o intersection o circle S ) and a line L 5 is given as
=+   LS   λ  , (where λ   is a parameter)
(-) The equation o the amily o circles touching the circle S ) and the line L 5 at their point o  contact P is
=+   LS   λ  , (where λ   is a parameter)
(/) The equation o a amily o circles passing
through two given points ),(   11  y  x P  and ),(   !!  y  x Q
can 'e written in the orm
)()()()(
!!
,
() The equation o amily o circles, which touch
)( 11   x  x m y  y    −=−  at ),( 11  y  x   or any fnite m is
)H()()(   1 !
1 !
1   y y y y x x   −+−+−   λ    )I( 1   =−−   x  x m  
And i m is infnite, the amily o circles is
)()()( 1 !
1 !
1   =−+−+−   x  x  y  y  x  x    λ  , (where λ    is
a parameter)
(6) Dquation o the circles given in diagram is
+−− )()( !1   x  x  x  x  )()H(cot)()(   !1!1   y y x x y y y y   −−±−−   θ 
)I()( 1!   =−−−   y  y  x  x 
%adical axis
 The radical axis o two circles is the locus o a point which moves such that the lengths o the tangents drawn rom it to the two circles are equal.
 The equation o the radical axis o the two circle is
!1   =−SS   i.e.,
straight line.
Properties of the radical ais
(i) The radical axis and common chord are identical or two intersecting circles.
(ii) The radical axis is perpendicular to the straight
line which oins the centres o the circles.
C 1
 
  Circle and System of Circles .
(iii) & two circles cut a third circle orthogonally, the radical axis o the two circles will pass through the centre o the third circle.
%adical centre
 The radical axes o three circles, ta+en in pairs, meet in a point, which is called their radical centre. 9et the three circles 'e
1 =S   .....(i), ! =S  .....(ii) and -  =S   .E.(iii)
9et the straight lines i.e., OL and O# meet in O. The equation o any straight line passing through O  is
)()( 1-!1 =−+−   SSSS   λ  , where λ is any constant.
4or 1=λ  , this equation 'ecome -!   =− SS ,
which is, equation o O.
 Thus the third radical axis also passes through the point where the straight lines OL and O# meet.
&n the a'ove fgure O is the radical centre.
Properties of radical centre
solving the equations -!1   SSS   == .
(ii) The radical centre o three circles descri'ed on the sides o a triangle as diameters is the orthocentre o  the triangle.
!o&axial s$stem of circles
A system (or a amily) o circles, every pair o which have the same radical axis, are called co0axial circles.
(1) The equation o a system o co0axial circles, when the equation o the radical axis and o one circle
o the system are =++≡   'my (x P ,
!!!! =++++≡   cfy gx  y  x S   respectively, is
=+   PS   λ    λ (  is an ar'itrary constant).
(!) The equation o a co0axial system o circles, where the equation o any two circles o the system are
!! 111 !!
and !! !!! !!
%espectively, is )( !11   =−+   SSS   λ 
or )( !11!   =−+   SSS   λ 
(-) The equation o a system o co0axial circles in
the simplest orm is !!! =+++   cgx  y  x  , where g  is a
varia'le and c is a constant.
Limiting points
9imiting points o a system o co0axial circles are the centres o the point circles 'elonging to the amily ("ircles whose radii are ero are called point circles).
9et the circle is !!! =+++   cgx  y  x  E..(i)
where g is a varia'le and c is a constant.
∴   "entre and the radius o (i) are ),(   g−  and
)(   ! cg   −   respectively. 9et ! =− cg   ⇒
cg   ±=
 Thus we get the two limiting points o the given co0
axial system as ),(   c  and ),(   c−
"learly the a'ove limiting points are real and distinct, real and coincident or imaginary according as c /, 5, J .
Image of the circle '$ the line mirror 
9et the circle 'e !!!! =++++   cfy gx  y  x   and line
mirror =++   'my (x  . &n this condition, radius o circle
remains unchanged 'ut centre changes. 9et the centre
o image circle 'e ),( 11   y  x  . *lope o × !1
CC  (slope
o 1)   −==++   'my (x  E..(i)
and mid point o ),(1   f gC   −−   and ),( 11!   y  x C
lies on =++   'my (x 
∴   %equired image circle is
S - 5
  / Circle and System of Circles
  A varia'le point moves in such a way that sum o 
sqhuare o distances rom the vertices o a triangle remains constant then its locus is a circle whose centre is the centroid o the triangle.    The reason why there are two equations
!1   mamx  y    +±=  o tangents is that there are two
tangents, 'oth are parallel and at the ends o a diameter.
   The line =++   cby ax   is a tangent to the circle
!!! r  y  x    =+  i and only i ).( !!!! bar c   +=    The condition that the line =++   'my (x 
touches the circle !!!! =++++   cfy gx  y  x   is
  ).()()( !!!!! cf gm('mf (g   −++=−+   Dquation o tangent to the circle
!!!! =++++   cfy gx  y  x    in terms o slope is
)1()(   !!! mcf gf mgmx  y    +−+±−+= .
   The length o the tangent drawn rom any point
on the circle !! 1 !! =++++   cfy gx  y  x   to the circle
!! !! =++++   cfy gx  y  x   is 1cc− .
  & two tangents drawn rom the origin to the circle
!!!! =++++   cfy gx  y  x   are perpendicular to each
other, then .!!! cf g   =+
  & the tangent to the circle !!! r  y  x    =+   at the
point (a, b) meets the coordinate axes at the points A and B  and O  is the origin, then the area o the
triangle OAB is ab
the circle !!! a y  x    =+  is     
 
 

 
 
−+ −
!!!
  & OA and OB are the tangents rom the origin to
the circle !! !! =++++   cfy gx  y  x  and C  is the
centre o the circle, then the area o the quadrilateral
OACB is .)(   !! cf gc   −+
! !!! =+++   cfy  y  x    touch each other, then
!!!
circle !!! )()(   ak  y h x    =−+− , then
)()( !!!! m(a'kmh(   +=++ .
  & O is the origin and OP, OQ are tangents to the
circle !! !! =++++   cfy gx  y  x  , then the circum0
centre o the triangle OPQ is      
      −−
! ,
!
!! !! =++++   cfy gx  y  x    'e r  and it touches 'oth
the axes, then r cf g   ===   :::: .
  9ength o an external common tangent and
internal common tangent to two circles is given 'y   9ength o external common tangent
! !1
tangent ! !1
where +  is the distance 'etween the centres o 
two circles i.e., + CC   =:: !1   and r 1  and r ! are the
radii o two circles.
  & the line cmx  y    +=   is a normal to the circle
with radius r  and centre at ),(   ba , then cmab   += .
  & θ  is the angle su'tended at ),(   11  y  x P 'y the
circle ,!!!! =++++=   cfy gx  y  x S   then
cf g
is cba /)( !
!!! r  y  x    =+   on the line 1=+ b
 y 
a
ba
babar  .
!!
!
the origin and the point ),(   f g  is !!
!!
!
!
+ =
and area o the triangle ormed 'y the pair o  tangents
and its chord o contact is !!
-
Fhere R is the radius o the circle and
L  is the length o tangent rom ),(   11   y  x P  on S5.
r 
1
 A
ere 1SL = .
  9ocus o mid point o a chord o a circle !!! a y  x    =+ , which su'tends an angle α   at the
centre is =+ !!  y  x    !)!8cos(   α a .
   The locus o mid point o chords o circle !!! a y  x    =+ , which are ma+ing right angle at centre
is !8!!! a y  x    =+ .
   The locus o mid point o chords o circle
!!!! =++++   cfy gx  y  x  , which are ma+ing right
angle at origin is . !
fy gx  y  x 
  & the points where the lines 111   =++   c y b x a
and !!!   =++   c y b x a  meet the coordinate axes are
con0cyclic then .!1!1   bbaa   =   & the equations o the circles whose radii are r 
and R 'e respectively =S and <=S , then the
equation o orthogonal circle is < =±
R
S
r 
S .