Find the equation of family of circles which are passing through the intersection of two circles S1: x2 + y2 - 8x - 2y + 7 = 0 and S2: x2 + y2 - 4x + 10y + 8 = 0 ?

CONCEPT:

The general second degree equation in x and y, a ⋅ x2 + 2hxy + b ⋅ y2 + 2gx + 2fy + c = 0 represents a circle with centre (-g, -f) and radius , when a = b and h = 0.

The equation of family of circles passing through the intersection of two circles S1 and S2 is given by: S1 + λ ⋅ S2 = 0 where λ ≠ 1.

Note: If λ = 1, then the equation S1 + λ ⋅ S2 will represent the equation of common chord of the two circles.

CALCULATION:

Given: S1: x2 + y2 - 8x - 2y + 7 = 0 and S2: x2 + y2 - 4x + 10y + 8 = 0 are two circles.

Here, we have to find the equation of the circle which passes through the intersection of the given circles S1 and S2 

As we know that, the equation of family of circles passing through the intersection of two circles S1 and S2 is given by: S1 + λ ⋅ S2 = 0 where λ ≠ 1.

Let's find out S1 + λ ⋅ S2 = 0

⇒ (x2 + y2 - 8x - 2y + 7) + λ ⋅ (x2 + y2 - 4x + 10y + 8) = 0

⇒ (1 + λ) x2 + (1 + λ) y2 - (8 + 4λ) x + (10λ - 2) y + (8λ + 7) = 0

So, the equation of the family of circles are: (1 + λ) x2 + (1 + λ) y2 - (8 + 4λ) x + (10λ - 2) y + (8λ + 7) = 0