Answer
See below
Work Step by Step
Given $-x^2-y^2-4x+12y+129=0$
We can see that $a=-1\\b=0\\c=-1$
We will find the discriminant of the given equation $=b^2-4ac\\=0^2-4(-1)(-1)\\=-1$
Since $-1 \lt 0$, the conic is a circle.
To graph the circle, first complete the square in x.
$-x^2-y^2-4x+12y+129=0\\-(x^2+4x+4)+4-(y^2-12y+36)-36=-129\\(x+2)^2+(y-6)^2=169$
From the equation, you can see that the center is at $(-2,6)$ and the radius is $r=13$
