Question

Prove that the centres of the three circles x² + y² – 4x – 6y – 12 = 0, x² + y²+ 2x + 4y – 5 = 0 and x² + y² – 10x – 16y + 7 = 0 are collinear.

Answer 1

Given,

x² + y² – 4x – 6y – 12 = 0

centre ( - g₁, - f₁) = (2, 3) 

x² + y² + 2x + 4y – 5 = 0

centre ( - g², - f²) = ( - 1, - 2) 

x² + y² – 10x – 16y + 7 = 0

centre ( - g₃, - f₃) = (5, 8)1

to prove that the centres are collinear,

= 2( - 2 - 8) - 3( - 1 - 5) + 1( - 8 + 10) 

= - 20 + 18 + 2 = 0

The centres are collinear.