When finding the closest and farthest distances between a circle and a line, I don't understand why the values I compute from the lagrange multipliers give both distances. I've attached a picture of my problem solving.
I understand why the normal line to the circle along the normal line to the line would be the shortest distance, but why is it also describing the maximum distance? according to the answers in the book my calculations are correct, but I don't understand geometrically why $\lambda = -4$ gives me the coordinates of the farthest point on the circle.