find the equation of the circle circumscribing the triangle formed by the lines x + y = 6, 2x + y = 4 and x + 2y = 5

Solving the given equation of the lines which form the sides of the triangle in pairs, the coordinates of the vertices of a triangle obtained are : (-2, 8), (1, 2) and (7, -1).

The equation of the circle be,

x² + y² + 2gx + 2fy + c = 0  ……………..(i)

Substituting the co-ordinates of the vertices obtained in equation (i), we get

-4g + 16f + c = -68  …………………..(ii)

2g + 4f + c =-5  …………………..(iii)

14g – 2f  + c = -50 …………………(iv)

By solving the equations (i), (ii) and (iii), we get

g= -17/2  ;  f = -19/2  and c = 50.

Now substituting the values of g, f and c in equation (i), we get

x² + y² – 17x – 19y + 50  = 0, which is the required equation of the circle.