find the minimum and maximum distance between both objects as function of $r$

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 $r$.

• Can you find the distance between the two objects? That would be a good start. – user88595 May 17 '14 at 15:37
• ya i found the distance but i don't understand "distance between both objects as function of r" – Rachel May 17 '14 at 16:00
• Btw, are you sure there are no typos in your formulas? They leads to rather hairy calculations. – Piotr Miś May 17 '14 at 16:36

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: $$L_{\max}(r) = \sqrt{F(t_{\max},r)} \qquad\qquad L_{\min}(r) = \sqrt{F(t_{\min},r)}$$
• 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. – Piotr Miś May 17 '14 at 16:44