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

Question

Philoid · Accepted Answer

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)

to prove that the centres are collinear,

Where x₁, y₁ are the coordinates of the ist centre and so on.

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

= - 20 + 18 + 2 = 0

The centres are collinear.

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.

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.