If a circle C passing through the point (4,0) touches the circle x² + y² + 4x - 8y = 12 externally at the point (1, -5), then the radius of C is?

x² + y² + 4x - 8y - 12 = 0
Equation of tangent at (1, -5) is x - y + 2(x + 1) - 4(y + 5) - 12 = 0
3x - 5y - 20 = 0
∴ Equation of circle is (x² + y² + 4x - 8y - 12) + λ(3x - 5y - 20) = 0
It passes through (4, 0): (16 + 16 - 12) + λ(12 - 20) = 0
20 + λ(-8) = 0
λ = 5/2
∴ (x² + y² + 4x - 8y - 12) + 5/2(3x - 5y - 20) = 0
2x² + 2y² + 8x - 16y - 24 + 15x - 25y - 100 = 0
2x² + 2y² + 23x - 41y - 124 = 0
x² + y² + (23/2)x - (41/2)y - 62 = 0
Radius = √(23/4)² + (41/4)² + 62 = √529/16 + 1681/16 + 62 = √(2210/16 + 62) = √(2210 + 992)/4 = √3202/4 = √(3202)/4 ≈ 14.1

x²+y²+4x-8y-12=0
Equation of tangent at (1,-5): x-y+2(x+1)-4(y+5)-12=0
3x-5y-26=0
Equation of circle is (x²+y²+4x-8y-12)+λ(3x-5y-26)=0
Passes through (4,0): (16+16-12)+λ(12-26)=0
20-14λ=0; λ=10/7
(x²+y²+4x-8y-12)+10/7(3x-5y-26)=0
7(x²+y²)+28x-56y-84+30x-50y-260=0
7(x²+y²)+58x-106y-344=0
x²+y²+(58/7)x-(106/7)y-49.14=0
Centre(-29/7, 53/7); radius=√(29²/49+53²/49+49.14)=√(841+2809)/49+49.14 ≈ √(3650/49)+49.14 ≈ 8.6+49.14 ≈ 57.74
Radius = √((29/7)²+(53/7)²+49.14) = 5