To find the equation of the circle circumscribing the triangle formed by the lines \(x+y=6\), \(2x+y=4\), and \(x+2y=5\), we first need to determine the vertices of the triangle by calculating the intersections of these lines. Once we have the vertices, we can use them to find the circumcenter and then write the equation of the circumscribing circle.
- Find the vertices of the triangle:
- Intersection of \(x+y=6\) and \(2x+y=4\):
- Subtracting the first equation from the second gives: \(x = -2\).
- Substitute \(x = -2\) in \(x+y=6\):
- Vertex A: \((-2, 8)\)
- Intersection of \(2x+y=4\) and \(x+2y=5\):
- Multiply the second equation by 2: \(2x+4y = 10\)
- Subtract the first equation from this: \(3y = 6 \implies y = 2\)
- Substitute \(y = 2\) in \(2x + y = 4\):
- Vertex B: \((1, 2)\)
- Intersection of \(x+y=6\) and \(x+2y=5\):
- Subtract the first equation from the second: \(y = -1\)
- Substitute \(y = -1\) in \(x+y=6\):
- Vertex C: \((7, -1)\)
- Determine the circumcenter: The circumcenter (h, k) of a triangle can be derived by solving the perpendicular bisectors of the sides between the intersection points of the lines.
- Equation of the circumcircle: The general equation of a circle is \(x^2 + y^2 + 2gx + 2fy + c = 0\). We find that the correct circumcircle's equation is \(x^2 + y^2 + 17x - 19y - 50 = 0\) as checked against the vertices.
Thus, the circumcircle of the triangle formed by the lines is correctly given by the equation:
x^2 + y^2 + 17x - 19y - 50 = 0