<s>[INST] Equation of Circle touching a straight line and passing through the centroid of a triangle and a particular point If the equation $3{x^2}y + 2x{y^2} - 18xy = 0$  represent the sides of a triangle whose centroid is G, then the equation of the circle  which touches the line $x-y+1=0$ and passing through G and $(1,1)$ is [/INST]  ${x^2} + {y^2} - x - y = 0$
B:  ${x^2} + {y^2} - 9x - y + 8 = 0$
C:  ${x^2} + {y^2} - 4x + 2y = 0$
D:  ${x^2} + {y^2} + 9x + y + 8 = 0$
I got the answer but had the choice not been provided how would I find the solution.
My steps are elaborated below
$3{x^2}y + 2x{y^2} - 18xy = 0 \Rightarrow xy\left( {3x + 2y - 18} \right) = 0$
The triangle is formed by line x=0, y=0 and 3x+2y-18=0
Centroid  $G = \left( {\frac{{6 + 0 + 0}}{3},\frac{{9 + 0 + 0}}{3}} \right) = \left( {2,3} \right)$
Let the centre of the circle is (h,k)
Circle passes through (2,3) and (1,1), hence the radius is
${r^2} = {\left( {h - 1} \right)^2} + {\left( {k - 1} \right)^2} = {\left( {h - 2} \right)^2} + {\left( {k - 3} \right)^2} \Rightarrow 2h + 4k = 11$
${x^2} + {y^2} + 2gx + 2fy + c + \lambda \left( {x - y + 1} \right) = 0$, represent the equation of the circle touching the line x-y+1=0 and passing through (2,3) and (1,1)
${x^2} + {y^2} + 2x\left( {g + \frac{\lambda }{2}} \right) + 2y\left( {f - \frac{\lambda }{2}} \right) + c + \lambda  = 0$
$Centre:\left( { - \left( {g - \frac{\lambda }{2}} \right), - \left( {f - \frac{\lambda }{2}} \right)} \right)$
Putting it in 2h+4k=1
$ - 2\left( {g - \frac{\lambda }{2}} \right) - 4\left( {f - \frac{\lambda }{2}} \right) = 11 \Rightarrow  - 2g - 4f + \lambda  + 2\lambda  = 11 \Rightarrow 2g + 4f = 3\lambda  - 11$
By Hit and Trial I am getting choice (B) as it satisfies $2h+4k=11$ where $h=\frac{9}{2}$ and $k=\frac{1}{2}$.
Had the option not been given how do I proceed
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