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Suppose that an object $O$ moves in the plane $x,y$ along a path with respect to time $ t $ of the form $O (t) = (x (t), y (t)) = (2 \cos (t) , 2 \sin (t))$ and another $ P $ object while moving along a path $ P (t) = (z (t), w (t)) = (r \cos (t + r ), \sin (t + r)) $ with $ t $ and $ r $ real numbers.

How can I find the minimum and maximum distance between both objects as function of $r$.

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    $\begingroup$ Can you find the distance between the two objects? That would be a good start. $\endgroup$ – user88595 May 17 '14 at 15:37
  • $\begingroup$ ya i found the distance but i don't understand "distance between both objects as function of r" $\endgroup$ – Rachel May 17 '14 at 16:00
  • $\begingroup$ Btw, are you sure there are no typos in your formulas? They leads to rather hairy calculations. $\endgroup$ – Piotr Miś May 17 '14 at 16:36
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I assume you mean Euclidean distance. Recall that the square of Euclidean distance between two points is given by: $$ F(t,r) = ||O-P||^2 = (x-z)^2 + (y-w)^2 = \dots $$ Now find extrema of $F$ treating $r$ as a constant parameter (simple calculus will do).

The minimal and maximal distance between objects can then be described as a function of r: $$ L_{\max}(r) = \sqrt{F(t_{\max},r)} \qquad\qquad L_{\min}(r) = \sqrt{F(t_{\min},r)} $$

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  • $\begingroup$ ya i found the distance but i don't understand "distance between both objects as function of r" $\endgroup$ – Rachel May 17 '14 at 15:59
  • $\begingroup$ @Knight, see edit. $\endgroup$ – Piotr Miś May 17 '14 at 16:35
  • $\begingroup$ but understand me i can find the minimum and max but i can't find the equation that i must minimize or maximize with that condition "distance between both objects as function of r", i found the distance but how put in parameter r $\endgroup$ – Rachel May 17 '14 at 16:39
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    $\begingroup$ Substitue formulas for $x$, $y$, $z$, $w$ into definition of $F$. You will get a function of two variables: $t$ and $r$. This is what you need to minimize/maximize. Treat $r$ as a constant and simply calculate derivative of $F$ with respect to $t$. When you are done, equate it to zero to find candidates for extremas. $\endgroup$ – Piotr Miś May 17 '14 at 16:44

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