The conversation between human and AI assistant.
In the xy-plane, what is the length of the shortest path from (0,0) to (12, 16) that does not go inside the circle (x -6) 2 + (y -8) 2 = 25?
Alright, how do we find the distance between the line segment and the circle?
We need to find the equation of the line that passes through (0,0) and (12,16). The equation of the line is y = (4/3)x. Then we use the formula for the distance from a point to a line. The formula is |Ax + By + C| / sqrt(A^2 + B^2), where A, B, and C are the coefficients of the equation of the line and (x,y) is a point on the circle. So the equation of the line is 3y = 4x, which means A = 4, B = -3, and C = 0. Since the center of the circle is at (6,8) and the radius is 5, we can use the point (6,8+5) = (6,13) to find the distance between the line and the circle. Plugging in the values, we get |4(6) - 3(13)| / sqrt(4^2 + (-3)^2) = 13/sqrt(25) = 13/5.