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# Find the equation of the circle which circumscribes the triangle formed by the lines:x + y = 2, 3x - 4y = 6 and x - y = 0

Given that we need to find the equation of the circle formed by the lines: x + y = 2

3x - 4y = 6

x - y = 0

On solving these lines we get the intersection points A(2,0), B(- 6, - 6), C(1,1)

We know that the standard form of the equation of a circle is given by:

x2 + y2 + 2ax + 2by + c = 0 .....(1)

Substituting (2,0) in (1), we get

22 + 02 + 2a(2) + 2b(0) + c = 0

4 + 4a + c = 0

4a + c + 4 = 0 ..... (2)

Substituting (- 6, - 6) in (1), we get

(- 6)2 + (- 6)2 + 2a(- 6) + 2b(- 6) + c = 0

36 + 36 - 12a - 12b + c = 0

12a + 12b - c - 72 = 0 ..... (3)

Substituting (1,1) in (1), we get

12 + 12 + 2a(1) + 2b(1) + c = 0

1 + 1 + 2a + 2b + c = 0

2a + 2b + c + 2 = 0 ..... (4)

Solving (2), (3), (4) we get

a = 2, b = 3,c = - 12.

Substituting these values in (1), we get

x2 + y2 + 2(2)x + 2(3)y - 12 = 0

x2 + y2 + 4x + 6y - 12 = 0

The equation of the circle is x2 + y2 + 4x + 6y - 12 = 0.

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