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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.

enter image description here

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Setting the gradient to zero finds extrema (or saddle points), not just max/min. The Lagrange multiplier method just takes the constraint into account, but will still not distinguish between min,max, saddle points. –  copper.hat Jun 14 '12 at 19:34
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It is partly an "accident," (or more precisely a property of the circle) that a line perpendicular to a tangent is also perpendicular to a tangent at the other intersection point. –  André Nicolas Jun 14 '12 at 19:47

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