8/7/2019 families of circles

http://slidepdf.com/reader/full/families-of-circles 1/10

Families of Circle

8/7/2019 families of circles

http://slidepdf.com/reader/full/families-of-circles 2/10

Equation of circle: (x-3)2 + (y -1)2 = k2

Center at the origin ( 3,1) the radius

varies

k = 3

k =

5

k = 8

k = 9

Concentric circles: the same center

point.

8/7/2019 families of circles

http://slidepdf.com/reader/full/families-of-circles 3/10

Ex. 1. Write an equation which represents a family of  

circles with center on the line y = x and which passes

through the origin. Find the member of this family which

passes through the point ( 6 ,0).

Ans. (x

- h )2

+ ( y - k )2

= r2

( x - 3 ) 2 + ( y - 3 )2 = 18

8/7/2019 families of circles

http://slidepdf.com/reader/full/families-of-circles 4/10

Circle Passing Through the Intersection

of Two Circles

Circle 1: x2 + y2 + D1

x + E1

y + F1

= 0

Circle 2: x2 + y2 + D2x + E2y + F2 = 0

(x2 + y2 + D1

x + E1

y + F1

) + k (x2 + y2 + D2

x + E2

y + F2

) = 0

Equation of a family of circles passing

through the intersection of two circles:

where k -1

8/7/2019 families of circles

http://slidepdf.com/reader/full/families-of-circles 5/10

Ex 1. Write the equation of the family of circles C3

all members of which pass

through the intersection of the circles C1 and C2 represented by the equations

C1: x2 + y2 + 2x - 4y - 4 = 0

C2: x2 + y2 - 6x + 2y - 6 = 0

Find the member of the family C3 that passes through the point ( 9, -1).

Answer:

Family: (x2 + y2 + 2x - 4y - 4 ) + k (x2 + y2 - -6x + 2y - 6 ) = 0

C3: k = -5; x2 + y2 - 8x + 7y/2 - 13/2 = 0

center ±radius form: ( x ± 4)2 + ( y + 7/4)2 = 409/16.

8/7/2019 families of circles

http://slidepdf.com/reader/full/families-of-circles 6/10

C1

C2

C3

C1: center: ( -1, 2) r = 3 

C2: center: ( 3, - 1) r = 4

C3: center: ( 4, -7/4) r =4

409

C1: C1: x2 + y2 + 2x - 4y - 4 = 0

C2: x2 + y2 - 6x + 2y - 6 = 0

k = - 4

8/7/2019 families of circles

http://slidepdf.com/reader/full/families-of-circles 7/10

The equation: (x2+ y2+ D1x + E1y + F1) + k (x2 + y2 + D2x + E2y + F2) = 0

represents a circle if k - 1

If k = - 1, equation of Circle 2 is subtracted from that of circle 1

and the equation becomes:

( D1 - D2) x + ( E1 - E2 ) y + F1 - F2 = 0

 Note: the above equation is linear so the graph of the equation is

a straight line which is the radical axis.

8/7/2019 families of circles

http://slidepdf.com/reader/full/families-of-circles 8/10

8/7/2019 families of circles

http://slidepdf.com/reader/full/families-of-circles 9/10

Ex., Find the equation of the radical axis of the circles

in the previous problem.

C1

C2

C1: x2 + y2 + 2x - 4y - 4 = 0

C2: x2 + y2 - 6x + 2y - 6 = 0

Ans. 4x - 3y + 1 = 0

8/7/2019 families of circles

http://slidepdf.com/reader/full/families-of-circles 10/10

Assignment:

1. Write the equation of the f amily of circles passing through the intersection

points of  

x2 +  y2 + 16x + 10y  +  24 = 0 and x2 +  y2 + 4x - 8y - 6 = 0

Find the member of the f amily that passes through the origin. Construct

the three circles. Ans. 5x2 + 5y2 +  32x - 22y = 0.

2. Write the equation of the f amily of circles passing through the intersection of  

two circles:x2 +  y2 +  2x + 4y - 4 = 0 and x2 +  y2 + 6x +  2y  + 6 = 0

Find the member for which k = 2. Ans. 3x2 +  3y2 + 14x +-8y  + 8 = 0.

or 3x2 +  3y2 + 10x + 10y - 2 = 0.

3. Draw the graph of the equations:

x2 +  y2 + 4x + 6y - 3 = 0 and x2 +  y2 + 12x + 14y  + 6 0 = 0. Then

find the equation of the radical axis and draw the axis. Ans. 8x + 8y = -63.