Combining like terms on the right hand side of the given equation gives $x^2 + y^2 - 7 = -12x - 2y - 11.$ Completing the square in $x$ and $y,$ we get
\[(x + 6)^2 + (y + 1)^2 = 92.\]Thus, the equation represents a circle with center $(-6,-1)$ and radius $\sqrt{92} = 2 \sqrt{23}.$
[asy]
unitsize(0.4 cm);
draw(Circle((-6,-1),3*sqrt(23)));
dot("$(-6,-1)$", (-6,-1), SW);
[/asy]
The area of the region defined by the given equation is thus $\boxed{92 \pi}.$
What is the area of the region defined by the equation $x^2+y^2 - 7 = 4y-14x+2x-6y+3$?
1 answer