### Instruction: What is the shortest distance from the origin to the circle defined by $x^2-24x +y^2+10y +160=0$?### Response: To find the shortest distance from the origin to the circle, I need to find the radius of the circle and the distance from the origin to the center of the circle.
To find the radius and the center, I can complete the square for both x and y terms in the equation of the circle.
I can rewrite the equation as $(x^2-24x+144) + (y^2+10y+25) = 9$, which is equivalent to $(x-12)^2 + (y+5)^2 = 3^2$.
This means the center of the circle is at $(12, -5)$ and the radius is 3.
The distance from the origin to the center is $\sqrt{12^2 + (-5)^2} = \sqrt{169} = 13$.
The shortest distance from the origin to the circle is then the difference between the distance to the center and the radius, which is $13 - 3 = 10$.