Centre

Important Elements

• Centre

• Radius

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 − 𝒄If coefficient of x2 and y2 are not 1 , then make them one by dividing

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

Tangent

• Condition for tangency :

•Distance of tangent from centre = Radius

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

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

• 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

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 :

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

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

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