We have to find the equation of a circunference with:
[tex]\begin{gathered} \text{Center}=(-3,7) \\ \text{Passes through the point P}=(-6,-2) \end{gathered}[/tex]
As it is a circunference that passes through a point, we know that the distance between the center and the point must be the radius, as all points in a circunference are at the same distance from the center.
We will find then the radius, by calculating the distance:
[tex]\begin{gathered} d(C,P)=\sqrt[]{(-3-(-6))^2+(7-(-2)_{})^2} \\ =\sqrt[]{(-3+6)^2+(7+2)^2} \\ =\sqrt[]{3^2+9^2} \\ =\sqrt[]{9+81} \\ =\sqrt[]{90} \end{gathered}[/tex]
Now, this means that the radius is √90.
We use the standard form for the equation of a circle:
[tex](x-h)^2+(y-k)^2=r^2[/tex]
where (h,k) is the center of the circle. In this case,
[tex](h,k)=(-3,7)[/tex]
And replacing, we obtain that the equation of the circle is:
[tex]\begin{gathered} (x+3)^2+(y-7)^2=(\sqrt[]{90})^2 \\ (x+3)^2+(y-7)^2=90 \end{gathered}[/tex]
