The document discusses important concepts related to circles such as the center, radius, equation of a circle, finding the center and radius from the standard form equation, writing the equation of a tangent line from a given point, finding the normal line from a given point, the equation of a chord of contact between a circle and tangent line, the radical axis between two circles, and the family of circles passing through the point of intersection between a circle and another circle or line. Examples are provided to illustrate each concept.

2. Centre
Important Elements
• Centre
• Radius

3. Equation of circle
Coefficient of x2 = Coefficient of y2
And coefficient of xy =0 , No xy term
General equation : x2 + y2 + 2gx + 2fy +c =0
Center = (−
𝑪𝒐𝒆𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕 𝒐𝒇 𝒙
𝟐
, -
𝑪𝒐𝒆𝒇𝒇𝒊𝒄𝒊𝒆𝒏𝒕 𝒐𝒇 𝒚
𝟐
)
Radius = 𝒈𝟐 + 𝒇𝟐 − 𝒄
If coefficient of x2 and y2 are not 1 , then make them one by
dividing

4. Example : x2 + y2 -4x +10y +4=0
Centre = 2, -5
Radius = root (4 +25 -4) =5
Example : 2x2 + 2y2 -8x +20y +8=0
Coefficent are not 1 , divide by 2
Dividing by 2 we get
x2 + y2 -4x +10y +4=0
Centre = 2, -5
Radius = root (4 +25 -4) =5

5. Tangent
• Condition for tangency :
• Distance of tangent from centre = Radius

6. Example : x2 + y2 -4x +10y +4=0 . Find tangent to
circle from (3,-4) , (0,0) and (-3,-5)
• Part 1
(3,-4) Centre = (2,-5) Radius =5
Distance of (3,-4) from centre (2,-5) = √2 < Radius
Point is inside the circle , No tangent can be drawn
Part 2 (0,0)
Distance from centre = √29 > Radius
Point is outside  Two tangents possible

7. Tangent from (0,0) will pass through (0,0)
So Equation can be written as (y-0) = m (x-0) => y = mx
For Tangency : Distance of y=mx from center = Radius
Distance of y=mx from (2,-5) = 5
|−5−2𝑚|
(1+𝑚2)
= 5 ( )
squaring both sides
4m2 + 20m + 25 = 25 (m2 +1)
21 m2 – 20m =0
m = 0 or m =20/21
Two tangents : y =0 and y =(20/21) x

8. • Part 3 : (-3,-5)
Distance from centre (2,-5 ) = 5 = Radius
Since Distance= radius  Point lies on circle  One tangent is possible
Any line from (-3,-5) can be written as (y+3) = m (x+3)
y –mx +3-3m =0
For this to be tangent : Distance from centre = Radius
|−5−2𝑚+3−3𝑚|
(1+𝑚2)
= 5
Square 25 m2+20m + 4 = 25 m2 + 25
m= 21/20 Only one value

9. Normal to circle
• Normal to circle passes through centre
• Normal is perpendicular to tangent at point of contact
Circle :x2 + y2 -4x +10y +4=0. Find normal from (0,0)
Normal will pass through (0,0 ) and centre (2,-5 )
Two points are known = > line is fixed
Use two point form :

10. Chord of contact
• AB is chord of contact in the figure
PA and PB are tangents
Tangent from same point are equal so Length(PA) = Length(PB)
• External point : (x1,y1)
• Circle : : x2 + y2 + 2gx + 2fy +c =0
To write equation of chord of contact
Replace x2 by x x1
Replace y2 by y y1
Replace x by (x + x1 )/2 Replace y by (y+y1)/2
So equation : x1x + y1y + g(x1 + x) + f(y1 +y) + c = 0

11. Radical axis
• Length of tangents drawn from any point on radical axis to circles is equal
• Circle 1 x2 + y2 + 2gx + 2fy +c =0 Circle 2 x2 + y2 + 2hx + 2ky +d =0
• Equation of radical axis = Circle 1 –Circle 2 =0

12. Family of circle
• Equation of circle passing through point of intersection of two circle
C1 and C2 : C1 + λ C 2 =0
• Equation of circle passing through point of intersection of circle C1
and Line L1 : C1 + λ L1 =0
• Line is Tangent  Distance from centre = R
• Line is secant  Distance from Centre < R
• Line is outside  Distance > R