1 General equation of a circle : x2 + y2 + 2gx + 2fy + c = 0 where g , f and c are constants
(i) center of the circle is (-g, -f)
(ii) Radius is Ö(g2 + f2 -c)
2 Central (Centre radius) form of circle :
(i) (x-h)2 + (y-k)2 = r2, where (h,k) is circle centre and r is the radius.
(ii) x2 + y2 = r2 , where (0,0) origin is circle center and r is the radius.
3 Diameter form : If (x1, y1) and (x2 , y2) are end pts. Of a diameter of a circle, then its equation is
(x–x1) (x-x2) + (y-y1)(y-y2) = 0
4 Parametric equation :
(i) the parametric equation of the cirde x2 + y2 = r2 are x = rcos q, y = rsin p where point q º (r cos q , r sin q)
(ii) the parametric equation of the circle (x-h)2 + (y-k)2 = r2 are x = h + rcos q , y= k + r sin q
(iii) the parametric equation of the circle x2 + v2 + 2ax + 2fv + c = 0 are
(iv) for circle x2 + y2 = a2 , equation of chord joining q1 & q2 is
5 Concentric circle : Two circle having same center C(h,k) but different radii r1 & r2 respectively are called concentric circle .
6 Position of a point w.r.t. a circle : A point (x1, y1) lies outside on or inside a circle
S º x2 + y2 + 2gx + 2ky + c =0 according as
S1 º x21 + y21 + 2gx1 + 2gx1 + 2fy1 + c is + ve, zero or- ve
7 Chord length : (length of intercept) =2 Ö(r2 -p2)
8 Intercepts made on coordinate axes by the circle :
(i) x axis = 2Ö(g2 -c)
(ii) y axis = 2Ö(f2 - c)
9 Length of tangent : Length of tangent = ÖS1
10 Length of the intercept mad by line : y = mx + c with the circle x2 + y2 = a2 is
Where |x1 - x2| = difference of roots i.e. q(D/a)
11 Condition of Tangency : Circle x2 + y2 = a2 will touch the line y = mx + c if c ¹ a Ö (1+m2)
12 Equation of tangent , T = 0:
(i) Equation of tangent to the circle x2 + y2 + 2gx + 2fy + c = 0 at any point (x1, y1) is xx1 + yy1 + g(x+x1) + f(y+y1) + c =0
(ii) Equation of tangent to the circle x2 + y2 = a2 at any point (x1, y1) is xx1 + yy1 = a2
(iii) In slope form : from the condition of tangency for every value of m.
The line y = mx ± a Ö(1+m2) is a tangent to the circle x2 + y2 = a2 and its point of contact is
(iv) Equation of tangent at (a cos q, a sin q) to the circle x2 + y2 = a2 is x cos q + y sin q = a
13. Equation of normal :
(i) Equation of normal to the circle x2 + y2 + 2gx + 2fy + c =0 at any point P(x1, y1) is
(ii) Equation of normal to the circle x2 + y2 = a2 at any point (x1, y1) is xy1 - x1y = 0
14 Equation of pair of tangents : Equation of pair of tangents SS1 = T2
15 The point of intersection : The point of intersection of tangent drawn to the circle x2 + y2 = r2 at point q1 & q2 is given as
16 Equation of the chord of contact :Equation of the chord of contact of the tangents drawn form point P outside the circle the is T =0
17 Equation of a chord whose middle :Equation of a chord whose middle pt. is given by T = S1
18 Director circle : Equation of director circle for x2 + y2 = a2 is x2 + y2 = 2a2 Director circle is a concentric circle whose radius is Ö2 times the radius of the given circle .
19 Equation of polar of point :Equation of polar of point (x1, y1) w.r.t. the circle S = 0 is T = 0
20 Coordinates of pole : coordinates of pole of the line |x + my + n = 0 w.r.t. the circle x2 + y2 = a2 are 
21 Family of Circles :
(i) S + lS’ = 0 represents a family of circle passing through the pts. Of intersection of S = 0 & S’ = 0 if l ¹ - 1
(ii) S + l L =0 represent a family of circle passing though the point of intersection of S = 0 & L = 0
(iii) Equation of circle which touches the given straight line L=0 at the given point (x1, y1) is given as (x - x1)2 + (y – y1)2 + l ¹ -1
(iv) Equation of circle passing through two point A (x1, y1) & B(x2. Y2) is given as
(x – x1) (x – x2) + (y – y1) (y – y2) + l 
22 Equation of common chord :Equation of common chord is S –S1 = 0
23 The angle of q of intersection: The angle of q of intersection of two circle with centres C1 & C2 and radii r1 & r2 is given by
24 Position of two circles : Let two circles with centres C1, C2 and radii r1, r2 .
Then following cases arise as
(i) C1 C2 > r1 + r2 Þ do not intersect or one outside the other 4 common tangents.
(ii) C1 C2 = r1 + r2 Þ Circle touch externally, 3 common tangents.
(iii) |r1 - r2| < C1 C2 < r1 + r2 Þ Intersection at two real points , 2 common tangents.
(iv) C1C2 = |r1 - r2| Þ internal touch, 1 common tangent.
(v) C1C2 < |r1 + r2| Þ one inside the other , no tangent
Note: point of contact divides C1 C2 in the ratio r1 : r2 internally or externally as the case may be
25 Equation of tangent at point :Equation of tangent at point of contact of circle is S1 - S2 =0
26 Radical axis and radical centre:
(i) Equation of radical axis is S – S1 = 0
(ii) The point of concurrency of the three radical axis of three circles taken in pairs is called radical centre of three circles.
27 Orthogonality Condition :
If two circles S º x2 + y2 + 2gx + 2fy + c = 0 and S’ = x2 + y2 + 2g’x + 2f’y + c’ = 0 intersect each other orthogonally, then 2gg’ + 2ff = c + c’.
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