Circle

Determine the coordinates of the center of the circle $k$ that passes through the points $A=[4; 5]$, $B=[2; 1]$, and $C=[-1; 0]$.

Tom solved this problem using the following steps:

(1) He substituted the coordinates of the given points into the general equation of the circle $k$: The general equation of the circle $k$ is $x^2 + y^2 + ax + by + c = 0$, where $a$, $b$, $c$ are constants and $[x; y]$ is an arbitrary point on the circle. \begin{aligned} A\in k: \ \qquad\quad 4^2 + 5^2 + a\cdot4 + b\cdot5 + c &= 0\cr B\in k: \ \qquad \quad 2^2 + 1^2 + a\cdot2 + b\cdot1 + c &= 0\cr C\in k:\, (-1)^2 + 0^2 + a\cdot(-1) + b\cdot0 + c &= 0 \end{aligned}

(2) After simplifying each of the equations, he wrote the system of three equations with unknowns $a$, $b$, and $c$, and found the system's solution: \begin{aligned} 41 + 4a + 5b + c &= 0\cr 5 + 2a + b + c &= 0\cr 1 - a + c &= 0\cr\hline \end{aligned}

From the third equation, Tom expressed $a = 1 + c$ and substituted it into the first two equations, which he then simplified. \begin{aligned} 41 + 4\cdot(1 + c) + 5b + c &= 0\cr 5 + 2\cdot(1 + c) + b + c &= 0\cr\hline \end{aligned} \begin{aligned} 45 + 5b + 5c &= 0\cr 7 + b + 3c &= 0\cr\hline \end{aligned} Now, from the second equation, Tom expressed $b = -7 - 3c$ and then substituted it into the first equation, which he then solved for $c$. \begin{aligned} 45 + 5\cdot(-7 - 3c) + 5c &= 0\cr 10 - 10c &= 0\cr c &= 1 \end{aligned}

Finally, Tom substituted the obtained value of $c$ into the relations for $a$ and $b$. \begin{aligned} a &= 1+c = 1 + 1 = 2\cr b &= -7-3c =-7-3\cdot1 =-10\cr \end{aligned}

(3) He wrote down the general form of the equation of the circle $k$: $$x^2 + y^2 + 2x - 10y + 1 = 0$$ and transformed it into the standard form: \begin{aligned} x^2 + 2x + y^2 - 10y + 1 &= 0 \cr \color{red}x^2 + 2x + 1\color{black} - 1 + \color{blue}y^2 - 10y + 25\color{black} - 25 + 1 &= 0\cr \color{red}(x + 1)^2\color{black} + \color{blue}(y - 5)^2\color{black} - 25 &= 0\quad / + 25\cr (x + 1)^2 + (y - 5)^2 &= 25\cr \end{aligned}

(4) Next, he used the standard form of the equation to determine the coordinates of the center of the circle $k$: $$S = [1; -5]$$

Is Tom's solution correct? If not, determine where Tom made a mistake.