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,

x^{2} + y^{2} + 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

x^{2} + y^{2} – 17x – 19y + 50 = 0, which is the required equation of the circle.

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