<s>[INST] What is the equation of the circle I've been experiencing a difficulties in answering this. I hope someone will help me in solving this
Find the equation of the circle through the points $(2,8),(7,3)$ and $(-2,0)$.
 [/INST] HINT
Let start from the general equation
$$(x-x_C)^2+(y-y_C)^2=R^2$$
and plug in the values $(x,y)$ for the given points to obtain three equations in the three uknowns $x_C$, $y_C$ and $R$.
We obtain the system

*

*$2x_1x_C+2y_1y_C-x_1^2-y_1^2=x_C^2+y_C^2-R^2$

*$2x_2x_C+2y_2y_C-x_2^2-y_2^2=x_C^2+y_C^2-R^2$

*$2x_3x_C+2y_3y_C-x_3^2-y_3^2=x_C^2+y_C^2-R^2$
that is

*

*$2x_1x_C+2y_1y_C-x_1^2-y_1^2=2x_2x_C+2y_2y_C-x_2^2-y_2^2$

*$2x_2x_C+2y_2y_C-x_2^2-y_2^2=2x_3x_C+2y_3y_C-x_3^2-y_3^2$
from which we can find $x_C$ and $y_C$ and then $R$.

We can check directly the solution $(x-2)^2+(y-3)^2=25$ you have obtaind as follows

*

*$(2,8) \to (2-2)^2+(8-3)^2=25$

*$(7,3) \to (2-7)^2+(3-3)^2=25$

*$(-2,0) \to (-2-2)^2+(0-3)^2=25$
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