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Problem 48 Medium Difficulty

If $ r = \langle x, y \rangle , r_1 = \langle x_1, y_1 \rangle $, and $ r_2 = \langle x_2, y_2 \rangle $, describe the set of all points $ (x, y) $ such that $ \mid r - r_1 \mid + \mid r - r_2 \mid = k $, where $ k > \mid r_1 - r_2 \mid $.

Answer

$\therefore$ the equation $\left|r-r_{1}\right|+\left|r-r_{2}\right|=k$ describes the set of all points $(x, y)$
outside of an ellipse with foci $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$

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Video Transcript

knowing here that um are equals X, Y, R one equals X and Y one and R two equals X two, Y two. We want to describe the set of all points such that the absolute value of R -R one, that's the absolute value of ar minus or two is equal to some constant K. So we see that this is going to be uh this is where K is greater than greater than uh the distance between R one and R two. So we see that we could write this as a distance. This is the distance formula, the square root, uh case x -11 Squared plus x -12 Squared. We could do all this with the distance from the but what we ultimately end up seeing as that this is going to describe all the points outside of an ellipse. So point outside of an ellipse, forgive and focus X Y back on.