Mathematics FAMILY OF CIRCLE

Family of Circles-Type 1

The equation of the family of circles passing through the points of intersection of two given circles `S = 0` and `S' = 0` is given as
`S +lambda S' = 0` (where `lambda` is a parameter, `lambda ne - 1`)

Illustration:
Find a circle passing through the intersection of `x^2 + y^2 - 4 = 0` and` x^2 + y^2 - 6x + 5 = 0` which passes through the point (2, 1)?

Solution:
Family of required circles is `S_1 + λ S_2 = 0`

⇒ `(x^2 + y^2 - 4) + λ (x^2 + y^2 - 6x + 5) = 0 `

Since the required circle passes through the point (2, 1), the previous equation is satisfied for the point (2, 1)

⇒ (4 + 1 - 4) + λ (4 + 1 - 12 + 5) = 0

`1 - 2λ = 0 ⇒ λ = ½`

∴ Equation of the required circle is

`(x^2 + y^2 - 4) + 1/2 (x^2 + 2y - 6x + 5) = 0`

`⇒ x^2 + y^2 - 2x - 1 = 0`

Family of Circles-Type 2

The equation of the family of circles passing through the points of intersection of circle `S = 0` and a line `L= 0` is given as
`S + lambda L = 0` (where `lambda` is parameter)

e.g. Find the equation of the circle described on the common chord of the circles `x^2 + y^2 – 4x – 5 = 0` and `x^2 + y^2 + 8y+ 7 = 0` as diameter

Solution:
Equation of the common chord is `S_1 – S_2 = 0`
`⇒ x + 2y + 3 = 0`

Equation of the circle through the two circles is `S_1 + λS_2 = 0`

⇒ `x_2 + y_2 -4/(1+λ ) x+8λy/(1+λ )+(7λ -5)/(1+λ ) = 0.`

Its centre `(2/(1+λ ),-4/(1+λ ))` lies on `x + 2y + 3 = 0`

`⇒ 2/(1+λ )-8/(1+λ ) + 3 = 0 ⇒ 2 – 8λ + 3 + 3λ = 0 ⇒ λ = 1.`

Hence the required circle is `x^2 + y^2 – 2x + 4y + 1 = 0.`

Family of Circles-Type 3

The equation of family of circles which touch `y - y_1 = m (x - x_1)`
at `(x_1, y_1)` for any finite `m` is
`(x - x_1)^2 + (y - y_1)^2+ lambda {(y - y_1) - m (x - x_1)} = 0`
and if `m` is infinite, the family of circles is
`(x - x_1)^2+ (y - y_1)^2 + lambda (x - x_1) = 0`
(where `lambda` is a parameter)

Family of Circles-Type 4

The equation of a fami ly of circles passing through two given
points `P(x_1, y_1)` and `Q(x_2, y_2)` can be written in the form
`(x - x_1)(x-x_2)+ (y - y_1)(y - y_2) + lambda |(x,y,1),(x_1,y_1, 1),(x_2 ,y_2,1)|=0`
(where `lambda` is a parameter)