What is the shortest distance between the circles defined by $x^2-24x +y^2-32y+384=0$ and $x^2+24x +y^2+32y+384=0$?
To find the shortest distance between two circles, I need to find the distance between their centers and subtract their radii. I can find the centers and radii of the circles by completing the square for both x and y in their equations. For the first circle, I have $x^2-24x +y^2-32y+384=0$. I can rewrite this as $(x-12)^2-144 +(y-16)^2-256+384=0$. Then I can simplify it to $(x-12)^2 +(y-16)^2=16$, which is the equation of a circle with center $(12,16)$ and radius $4$. For the second circle, I have $x^2+24x +y^2+32y+384=0$. I can rewrite this as $(x+12)^2-144 +(y+16)^2-256+384=0$. Then I can simplify it to $(x+12)^2 +(y+16)^2=16$, which is the equation of a circle with center $(-12,-16)$ and radius $4$. The distance between the centers of the circles is given by the Pythagorean theorem. I have $d=\sqrt{(12-(-12))^2 +(16-(-16))^2}$. Simplifying, I get $d=\sqrt{576+1024}=\sqrt{1600}=40$. The shortest distance between the circles is then $d-2r=40-2(4)=32$.

