Chinese Math Olympiad Question | Find the shortest distance from a circle to the origin

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Given a equation of a circle, can you find the shortest distance from this circle to the origin? There are different methods to solve this problem. For example, we can use calculus or trigonometric substitution to solve this question. In this video, we'll use a geometric method to solve it, which is easer. This is a question of Mathematical Olympiad.

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you mentioned trig substitution in the beginning, which is an approach i like.
Let x + 5 = 14cos(t), y - 12 = 14sin(t), therefore we wish to minimize (14cos(t) - 5)^2 + (14sin(t) + 12)^2.
This becomes 196 + 169 + 338sin(t) - 140cos(t). There is some angle alpha such that 338sin(t) - 140cos(t) = sqrt(338^2 + 140^2)sin(t + alpha) which would then achieve its minimum at -sqrt(338^2 + 140^2) = -364.
So the answer is 196 + 169 - 364 = 1.

prathikkannan
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Great problem 😊 But I must confess that I cheated - I used Lagrange multipliers to solve it 😂 Certainly not allowed in a Math Olympiad…

florianbuerzle