What is the distance between the center of the circle with equation $x^2+y^2=-4x+6y-12$ and the point $(1,7)$?
To find the distance between the center of the circle and the point, I need to know the coordinates of the center of the circle. I can find the center of the circle by completing the square for both $x$ and $y$ in the equation of the circle. I rewrite the equation as $(x^2+4x)+(y^2-6y)=-12$. To complete the square for $x$, I need to add $(4/2)^2=4$ to both sides of the equation. To complete the square for $y$, I need to add $(-6/2)^2=9$ to both sides of the equation. I get $(x^2+4x+4)+(y^2-6y+9)=-12+4+9$. I simplify the equation to $(x+2)^2+(y-3)^2=1$. This is the equation of a circle with center $(-2,3)$ and radius $1$. Now I can use the distance formula to find the distance between the center $(-2,3)$ and the point $(1,7)$. The distance formula is $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. I plug in the coordinates and simplify. I get $d=\sqrt{(1-(-2))^2+(7-3)^2}$. I get $d=\sqrt{3^2+4^2}$. I get $d=\sqrt{25}$. I get $d=5$. 